effect of frame shape and geometry on the global by …

EFFECT OF FRAME SHAPE AND GEOMETRY ON THE GLOBAL

BEHAVIOR OF RIGID AND HYBRID FRAME UNDER

EARTHQUAKE EXCITATIONS

by

S. M. ASHFAQUL HOQ

Presented to the Faculty of the Graduate School of

The University of Texas at Arlington in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING

THE UNIVERSITY OF TEXAS AT ARLINGTON

May 2010

iii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my thesis advisor, Dr. Ali

Abolmaali, for his continuous help and guidance in the accomplishment of this work. I

really appreciate Dr. Ali Abolmaali for his unrestricted personal guidance throughout this

study, for bringing out the best of my ability. His kind supervision, encouragement,

assistance and invaluable suggestion at all stages of the work made it possible to

complete this work. The services he offered personally were a great help.

I would like to express my sincere thanks to Dr. John H. Matthys and Dr.

Guillermo Ramirez, for their time to serve on my thesis committee and providing

invaluable suggestions and advice.

I would like to thank Dr. Tri Le, Dobrinka Radulova, Mohammad Razavi and all

my colleagues and friends for their moral support and helpful discussions.

Finally, I would like to express my heartfelt thanks to my parents and siblings for

their unconditional love, support, encouragement and blessings to complete my masters

program.

April 15, 2010

iv

ABSTRACT

EFFECT OF FRAME SHAPE AND GEOMETRY ON THE GLOBAL

BEHAVIOR OF RIGID AND HYBRID FRAME UNDER

EARTHQUAKE EXCITATIONS

S.M.Ashfaqul Hoq, M.S.

The University of Texas at Arlington, 2010

Supervising Professor: Dr. Ali Abolmaali

The primary focus of this study is to present a shape which will improve building

performance in earthquake excitations. The investigation started with a 3-, 9-, 12- and 20-

story rectangular frame. The seismic performance is observed for different earthquake

data with different frequency content. Then, the Rhombus Shape is introduced to

compare with the Rectangular Shape frames, keeping the height-to-width ratio and

loading the same. For the first part of the investigation, deflection and member forces are

compared between rhombus and rectangular shape considering all connections rigid. In

the second part of the study partially restrained connections are introduced in mid levels

of the high rise frames to observe the redistribution pattern of internal forces. The results

show that with all rigid connections, a rhombus shape performs better than a rectangular

frame in most of the cases. Also, partially restrained connections which forms hybrid

frame have more significant effects on the rectangular frame than the rhombus frame.

v

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ............................................................................................... iii

ABSTRACT ....................................................................................................................... iv

LIST OF ILLUSTRATIONS ............................................................................................. ix

LIST OF TABLES ............................................................................................................ xii

Chapter Page

1. INTRODUCTION .................................................................................................1

1.1 Introduction .................................................................................................1

1.2 Background .................................................................................................6

1.2.1 Moment Frames (MFs) ........................................................................6

1.2.2 Concentrically Braced Frames (CBFs) ................................................8

1.2.3 Eccentrically Braced Frames (EBFs) .................................................10

1.3 Literature review .......................................................................................11

1.4 Objective of the Present Study ..................................................................16

1.5 Scope of the Study ....................................................................................16

1.7 Organization of the Present Study ............................................................17

2. BACKGROUND ON FRAME ANALYSIS .......................................................18

2.1 Introduction ..............................................................................................18

2.2 Analysis Methods.....................................................................................18

vi

2.2.1 Elastic Analysis Methods ..............................................................19

2.2.2 Inelastic Analysis Methods ..........................................................20

2.3 First Order Elastic Analysis .....................................................................20

2.4 Second Order Elastic Analysis.................................................................21

2.5 Inelastic Analysis .....................................................................................23

2.6 Dynamic Analysis of frame .....................................................................25

2.7 Modal Analysis ........................................................................................27

2.8 Step-by-Step Integration ..........................................................................28

2.8.1 Newmark’s Method .....................................................................29

2.8.1.1 Average Acceleration Method ..........................................30

2.8.1.2 Linear Acceleration Method .............................................31

2.8.2 Wilson θ Method ..........................................................................32

2.8.3 Hilber-Hughes-Taylor Method ....................................................33

3. EFFECT OF FRAME SHAPE AND GEOMETRY ON SEISMIC PERFORMANCE ........................................................................34

3.1 Introduction ..............................................................................................34

3.2 Three Story Frame ...................................................................................36

3.3 Nine Story Frame .....................................................................................38

3.4 Twelve Story Frame .................................................................................41

3.5 Twenty Story Frame ................................................................................43

3.6 Eathquake in consideration ......................................................................46

3.6.1 Generated Earthquake ..................................................................47

vii

3.6.2 ElCentro .......................................................................................51

3.6.3 Northridge ....................................................................................53

3.6.4 Parkfield .......................................................................................55

3.6.5 Kocaeli .........................................................................................57

3.7 Discussion of the Results .........................................................................61

4. HYBRID FRAME SYSTEMS ...........................................................................67

4.1 Introduction ..............................................................................................67

4.2 Semi-rigid Connection .............................................................................69

4.2.1 Moment-Rotation models ............................................................70

4.2.2 Bi-linear semi-rigid Connection Properties .................................72

4.3 Two-Story Test Frame .............................................................................73

4.4 20-Story Rectangular Hybrid Frame ........................................................76

4.5 20-Story Rhombus Hybrid Frame ............................................................79

4.6 Discussion of the Results .........................................................................82

5. CONCLUSIONS AND RECOMMENDATIONS .............................................92

5.1 Summary ..................................................................................................92

5.2 Conclusions ..............................................................................................93

5.3 Recommendations ....................................................................................95

viii

APPENDIX

A. TOP LATERAL DISPLACEMENT .........................................................96

B. AXIAL FORCES IN THE FRAME ELEMENTS ..................................112

C. SHEAR FORCES IN THE FRAME ELEMENTS ..................................153

D. BENDING MOMENTS IN THE FRAME ELEMENTS ........................194

E. MAXIMUM LATERAL DISPLACEMENT PROFILE .........................235

F. LATERAL INTER STORY DRIFT ........................................................251

G. LOAD CALCULATION .........................................................................267

REFERENCES ................................................................................................................270

BIOGRAPHICAL INFORMATION ...............................................................................276

ix

LIST OF ILLUSTRATIONS

Figure Page

1.1 20-Story Rectangular frame and equivalent Rhombus shape ........................................4

1.2 Typical CBF configurations ...........................................................................................9

1.3 Typical EBF configurations .........................................................................................11

2.1 Generalised Load-Displacement curve for different types of analysis ........................21

2.2 P-δ and P-Δ effects.......................................................................................................22

2.3 Average acceleration ....................................................................................................30

2.4 Linear acceleration .......................................................................................................31

2.5 Linear variation of acceleration over extended time steps ...........................................32

3.1 Three-Story Rectangular and Rhombus frame ............................................................37

3.2 First two Mode Shapes for 3-Story Rectangular Frame (a) Mode 1, (b) Mode 2 ................................................................................................37

3.3 First two Mode Shapes for 3-Story Rhombus Frame (a) Mode 1, (b) Mode 2 ................................................................................................37

3.4 Nine-Story Rectangular and Rhombus frame ..............................................................39



3.7 Twelve-Story Rectangular and Rhombus frame ..........................................................41


x


3.10 Twenty-Story Rectangular and Rhombus frames ......................................................44

3.11 First two Mode Shapes for 20-Story Rectangular Frame (a) Mode 1, (b) Mode 2 ..............................................................................................45

3.12 First two Mode Shapes for 20-Story Rhombus Frame (a) Mode 1, (b) Mode 2 ..............................................................................................45

3.13 Acceleration Time History of Generated Earthquake ................................................48

3.14 Fourier Spectrum of Generated Earthquake ..............................................................49

3.15 20S-289_100_2.89-TDux(RMR,RCR)-generated data .............................................50

3.16 Acceleration Time History of El Centro Earthquake .................................................52

3.17 Fourier Spectrum of El Centro Earthquake ...............................................................52

3.18 Acceleration Time History of Northridge Earthquake ...............................................54

3.19 Fourier Spectrum of Northridge Earthquake .............................................................54

3.20 Acceleration Time History of Parkfield Earthquake .................................................56

3.21 Fourier Spectrum of Parkfield Earthquake ................................................................56

3.22 Acceleration Time History of Kocaeli[Afyon Bay,N(ERD)] Earthquake .................58

3.23 Fourier Spectrum of Kocaeli[Afyon Bay,N(ERD)] Earthquake ................................58

3.24 Acceleration Time History of Kocaeli[Aydin,S(ERD)] Earthquake .........................59

3.25 Fourier Spectrum of Kocaeli[Aydin,S(ERD)] Earthquake ........................................60

4.1 Rotational deformation of a connection due to flexure ...............................................70

4.2 Typical moment-rotation curves ..................................................................................71

4.3 Semi-rigid element of a Hybrid frame .........................................................................71

4.4 Bi-linear moment-rotation curve ..................................................................................72

xi

4.5 Two story one bay test frame .......................................................................................74

4.6 Moment-Rotation relation used in previous study (Bhatti and Hingtgen,1995) ..........75

4.7 Hybrid Rectangular: 20S – 289_100_2.89 – (RCSR) – E(7-11) .................................78

4.8 Hybrid Rhombus: 20S – 289_100_2.89 – (RMSR) – E(7-11) ....................................80

4.9 Fat Hybrid Rhombus: 20S – 289_300_0.963 – (RMSR) – E(7-11) ............................81

4.10 Maximum lateral sway 20S-289_100_2.89–MDUx (RCR,RCSR)–E1m(7-11) .......82

4.11 20S – 289_100_2.89–LDr (RCR,RCSR) –E1m(7-11) ..............................................84

4.12 20S – 289_100_2.89 – BM (RCSR/RCR) – E1m(7-11) for Beams ..........................85

4.13 Maximum lateral sway 20S-289_100_2.89-MDUx(RMR,RMSR)-E1m(7-11) ........88

4.14 20S –289_100_2.89–LDr (RMR,RMSR)–E1m(7-11) ..............................................88

4.15 20S – 289_100_2.89 – BM (RMSR/RMR) – E1m(7-11) for Beams ........................89

xii

LIST OF TABLES

Table Page

3.1 Modal period and frequencies for 3-Story Frames (SAP 2000) ..................................38

3.2 Modal period and frequencies for 9-Story Frames (SAP 2000) ..................................40

3.3 Modal period and frequencies for 12-Story Frames (SAP 2000) ................................43

3.4 Modal period and frequencies for 20-Story Frames (SAP 2000) ................................46

3.5 Frequency Content for El Centro Earthquake ..............................................................53

3.6 Frequency Content for Northridge Earthquake ............................................................55

3.7 Frequency Content for Parkfield Earthquake ..............................................................57

3.8 Frequency Content for Kocaeli[Afyon Bay,N(ERD)] Earthquake ..............................59

3.9 Frequency Content for Kocaeli[Aydin,S(ERD)] Earthquake ......................................60

3.10 Dominant frequency order for the considered earthquake .........................................61

4.1 Semi-rigid Connection properties ................................................................................73

4.2 Lateral displacement (inch) for 2-Story test frame ......................................................75

4.3 Absolute Maximum Bending Moment (kip-inch) for 2-Story test frame ....................76

4.4 Rectangular rigid and hybrid frame frequencies (Opensees) .......................................83

4.5 Rhombus rigid and hybrid frame frequencies (Opensees) ...........................................87

1

CHAPTER 1

INTRODUCTION

1.1 Introduction

Among different loads, earthquake is the most uncertain in nature. It develops

under earth due to movement of the plate tectonics and that is the reason it cannot be

predicted when and with how much energy it will be generated. Like other loads, it

cannot be calculated precisely or forecasted with reasonable accuracy. Other than the

earthquake, the structural response which is dynamic in nature is also quite unpredictable

due to the variable associated such as soil properties, material and geometry of the

structure, construction quality, location of the structure, magnitude and frequency of

ground motion, epicenter, focal depth, etc. Due to the unpredictable nature of the load it

has always been a challenging task for structural researchers and professionals to prepare

any guideline for earthquake engineering.

Structural steel framing system evolved in an attempt to resist lateral forces. After

1906, it was highlighted that the buildings with steel frame performs better in earthquakes

than the masonry structures (FEMA-355e). Prior to the San Francisco earthquake at this

year there was no provision for earthquake in the building codes. After Santa Barbara

earthquake in 1925, Uniform Building Code (UBC) was the first to include seismic

provisions in 1927. In 1959 Structural Engineers Association of California (SEAOC)

2

issued “lateral force recommendation” which was later adopted by the UBC in 1961. The

key feature was the requirement of steel moment resisting frame for tall buildings over

160ft. American Institute of Steel Construction (AISC) includes seismic provisions in

their 1992 specifications. With each major earthquake the building code is modified

(FEMA-355e). At this point, it was believed that the steel moment resisting frames are

adequate to dissipate earthquake energy with the presumed ductile behavior of its

moment resisting connections. But, the Northridge earthquake in 1994 challenged this

assumption when many beam-column connections failed in a brittle manner. This type of

brittle failure causes little observable damage which raises the concern about

undiscovered damage in past earthquakes. After Northridge, investigation has confirmed

such type of damage in some buildings subjected to Loma Prieta (1989), Landers (1992),

and Big Bear (1992) earthquake (FEMA-355f).

The significant amount of monetary losses in the Northridge and Loma Prieta

earthquakes triggered the popularity of the performance based seismic design. Vision

2000 report by the SEAOC (1995) highlighted the fact of economic losses even in the

moderate earthquakes. It was identified the need of a design and construction procedure

which could control the damage to acceptable limits. These limits were same as those

described in the FEMA-273 developed by the Applied Technology Council (ATC). The

four performance levels labeled as Operational, Immediate Occupancy, Life Safety and

Collapse Prevention were the state of the defined and observable damage in the structure.

3

Federal Emergency Management Agency published the FEMA-355f prepared by

the SAC joint venture in September 2000, where it narrated performance prediction and

evaluation technique for moment resisting frames along with the seismic hazard levels

and analysis procedures. An important feature of this procedure was to state capacity and

demand in terms of story drift. It should be mentioned that prior to the 1976

specifications there was no limits for lateral drift in seismic design.

FEMA-356 tabulated some typical drift values to describe the overall structural

response. A steel moment frame with 5% transient or permanent drift will fall in the

collapse prevention level. For life safety it was 2.5% transient and 1% permanent drift;

for immediate occupancy it was 0.7% transient with negligible permanent drift. For

braced steel frame these drift values were 2% transient or permanent, 1.5% transient and

0.5% permanent; 0.5% transient with negligible permanent for collapse prevention, life

safety and immediate occupancy level respectively. But, these values were not drift limits

requirements. Vision 2000 by the SEAOC imposed some drift limitations for steel

moment frames as 2.5% transient or permanent drift for collapse prevention, 1.5%

permanent and 0.5% transient for life safety level and 0.5% transient with no permanent

drift for operational level. Also, the connection requirements according to the Seismic

Provisions of AISC 2005 requires that beam to column connections be able to carry

minimum 0.04 radians of inter story drift angle.

4

To keep the inter story drifts within the specified limits a rhombus shape was

proposed as a seismic load resisting system in the first phase of this study instead of the

popular rectilinear assemblage of beams and columns. The proposed shape is created

from the rectangular moment frame by placing a rhombus with the same height and width

as the rectangle frame. Then the pure rhombus shape is generated by removing all the

members outside of it. A proposed 20-story rhombus frame with the corresponding

rectangular frame is shown in Figure 1.1.

Figure 1.1 20-Story Rectangular frame and equivalent Rhombus shape

5

The idea behind this shape is to utilize the advantage of both the rectangular

moment frames and concentrically braced frames. The first type of framing system uses

the flexural stiffness of the members to gain lateral stiffness; while in the later type,

internal axial stiffness of the diagonals are the main sources for lateral stiffness. Different

framing systems are discussed briefly in the background section of this chapter.

To study the performance of the proposed rhombus shape, a wide range of steel

frames are selected with different height-to-width ratios. These frames represent low-rise

to high-rise buildings. Rhombus shape is compared with rectangular shape under

different earthquake excitations. As low-rise buildings are usually vulnerable to high

frequency earthquake and high-rise buildings are vulnerable to low frequency earthquake,

four different earthquakes are selected with wide range of frequency content. These are

Elcentro (1940), Northridge (1994), Parkfield (1966) and two data for Kocaeli (1999)

earthquake.

In the second phase of this study the concept of eccentrically braced frame (EBF)

is presented. The energy dissipation characteristic of the “link” of an EBF system is tried

to be incorporated in the frame system in the form of semi-rigid connections. In this part

of the study hybrid frames with rigid and semi-rigid connections are analyzed. The effect

of semi-rigid connections on global behavior of the hybrid rectangular and rhombus

frames are compared with the rigid frames. The eccentrically braced frame system and

the semi-rigid connections are briefly discussed in the background section.

6

1.2 Background

1.2.1 Moment Frames (MFs)

Moment resisting frames are the rectangular assemblage of beams and columns.

The beams are rigidly connected to the columns. Primarily the rigid frame action

provides resistance to lateral forces which develops bending moments and shear forces in

the frame members and joints. A moment frame cannot displace laterally without bending

the beams and columns due to its rigid beam-column connections. The bending rigidity

and strength of the frame members provide the lateral stiffness and strength for the entire

structure. For several reasons steel moment resisting frames have been popular in high

seismic regions. It is viewed as a highly ductile system among all the structural systems.

Also, large force reduction factors are assigned to design earthquake forces in building

codes. No bracing members are present to block the wall openings which provide

architectural versatility for space utilization. But, compared to other braced systems

moment frames generally required larger member sizes than those required only for

strength alone to keep the lateral deflection within code approved drift limits. Again, the

inherent flexibility of the system may introduce drift-induced nonstructural damage under

earthquake excitation than with other stiffer braced systems. Even these perceptions

regarding the expected performance of steel moment frames in energy dissipation under

lateral loads was sacrificed after the 1994 Northridge earthquake when the steel moment

frames did not perform as expected. Brittle failures occurred at beam-column connections

which challenge the assumption of high ductility of the system (Michel Bruneau et al.

1998).

7

Moment frames are composed of beam, column, and panel zone. Panel zone is the

portion of the column contained within the joint region at the intersection of beam and

column. In traditional analysis moment frames are often modeled with dimensionless

nodes which are the intersection of beam and column members. Such models do not

consider panel zone. But ductile moment frames require explicit consideration of panel

zone. Depending on the yield strength and the yield thresholds, the beam, column and

even panel zone could contribute to the total plastic deformation at the joint. A structural

component considerably weaker than the other framing into the joint will have to provide

the needed plastic energy dissipation. Those structural components expected to dissipate

hysteretic energy during an earthquake must be detailed to allow the development of

large plastic rotations. Plastic rotation demand is typically obtained by inelastic response

history analysis. Without considering panel zone plastic deformations it was expected

that the largest plastic rotations in the beams are 0.02 radian (Tsai 1988, Popov and Tsai

1989). After the Northridge earthquake the required connection plastic rotation capacity

was increased to 0.03 radian for new construction and for post earthquake modification of

existing building it was 0.025 radian (SAC1995b).

8

1.2.2 Concentrically Braced Frames (CBFs)

The concentrically braced frame is a lateral force resisting system. It is an

efficient frame system marked by its high elastic stiffness which is commonly used to

resist wind or earthquake loadings. The system achieves high stiffness by its diagonal

bracing members which resist lateral forces by using higher internal axial actions and

relatively lower flexural actions. Diagonal bracings form the main units which provide

lateral stiffness in a CBF system. Braces can be in the form of I-shaped sections, circular

or rectangular tubes, double angle attached together to form a T-shaped section, solid T-

shaped sections, single angles, channels and tension only rods and angles. Bracing

members are commonly connected with other members of the framing system by welded

or bolted gusset plates. The CBF design method generally focuses on dissipating energy

in the braces such that the connection is designed to remain elastic at all times. To

maximize the energy dissipation, the brace connections should be designed to be stronger

than the bracing members they connect so that the bracing member can yield and buckle

(Michel Bruneau et al. 1998). Some common CBF systems are shown in Figure 1.2.

9

Figure 1.2 Typical CBF configurations

CBF systems are considered to be less ductile seismic resistant structure as

compared to other systems due to failure of the bracing members under large cyclic

displacements. These structures can experience large story drift after buckling of bracing

members, which in turn may lead to fracture of bracing members. Recent analytical

studies have shown that CBF system designed by conventional elastic design method can

undergo severe damage under design level ground motions (Sabelli, 2000). Current

seismic codes (ANSI, 2005a) has provisions to design ductile CBF which is also known

as Special Concentrically Braced Frames (SCBFs).

10

1.2.3 Eccentrically Braced Frames (EBFs)

The eccentrically braced frame is a combination of concentrically braced frame

and moment resisting frame. The EBF combines individual advantages of each frame and

minimizes their respective disadvantages. It possesses high elastic stiffness, stable

inelastic response under cyclic lateral loading and excellent ductility and energy

dissipation capacity (Michel Bruneau et al. 1998). Eccentric braced frames utilize both

the axial loading of braces and flexure of beam sections to resist the lateral forces. It

addresses the need for a laterally stiff framing system with large energy dissipation

capabilities under large seismic forces. The key distinguishing feature of an EBF is the

isolated segment of a beam termed as “link.” A typical EBF consists of a beam, one or

two braces and columns. Its configuration is similar to other conventional braced frames

with the exception that each brace must be eccentrically connected to the frame.

Eccentric connection introduces shear and bending in the beam adjacent to brace. The

short segment of the frame where those forces are concentrated is the “link.” All inelastic

activity is intended to be confined to the properly detailed link. Links act as structural

fuses which dissipate seismic input energy without much degradation of strength and

stiffness and thereby transfer less force to the adjacent columns, beams and braces.

Common EBF arrangements are given in Figure 1.3.

11

Figure 1.3 Typical EBF configurations

Lateral stiffness of the EBF is a function of the ratio of link length to the beam

length. As the link becomes smaller the frame becomes stiffer approaches the stiffness of

CBF. And as the link becomes longer the frame becomes more ductile approaching to the

stiffness of a moment frame.

1.3 Literature review

PRE-NORTHRIDGE DESIGN

The presumed ductility of the discussed framing systems are dependent on the

connection properties. The welded moment connections were popular in North American

seismic regions before Northridge earthquake. By the 1960s the building industry was

frequently using an alternative connection detail with a bolted web connection and fully

welded flanges. To see the plastic behavior of these moment connections the first test was

conducted in 1960s. Popov and Pinkney (1969) tested 24 beam column joints.

Specimens with welded flanges and bolted connections showed superior inelastic

12

behavior compared with the cover plated moment connection and the fully bolted

moment connection due to the fact of slippage of the bolts which causes visible pinching

of the hysteresis loops under cyclic loading (FEMA-355e) (Michel Bruneau et al. 1998).

In 1970s, welded flanges-bolted web connections with fully welded connections

are compared (Popov and Stephen 1970) and the fully welded connection exhibited more

ductile behavior. Four out of five bolted webs failed abruptly. Popov and Stephen (1972)

also concluded that “The quality of workmanship and inspection is exceedingly important

for the achievement of best results (Michel Bruneau et al. 1998).”

Popov et al. (1985) tested eight specimens. The test mainly emphasize on panel

zone behavior with W18 beams. As per the authors, during the welding procedure: “the

back-up plates for the welds on the beam flange-to-column flange connections were

removed after the full-penetration flange welding was completed and small cosmetic

welds appeared to have been added and ground off on the underside (FEMA-355e).”

The tests by Tsai and Popov (1987) and Tsai and Popov (1988) indicated some

prequalified moment connections in ductile moment frames with W18 and W21 similar

in depth to those tested by Popov and Stephen (1971), were not as ductile as expected.

Before developing adequate plastic rotations, specimens with welded flanges-bolted web

connections failed abruptly. Only four out of eight specimens achieve desirable beam

13

plastic rotation. Authors realized about the quality control as an important factor (FEMA-

355e)

Engelhardt and Husain (1993) conducted 8 tests to investigate the effect of the

ratio of Zf /Z on rotation capacity using W21 and W24 beams, where Zf is the plastic

modulus of the beam flanges and Z is the plastic modulus of the entire beam. They could

not find any relation between Zf /Z with amount of hysteretic behavior developed prior to

failure. But, interestingly some of the specimens showed lack of ductility. This study also

compared their results with past experimental data. Assuming that connections must have

a beam plastic rotation capacity of 0.015 radian to survive under severe earthquake, they

found none of the specimens could provide that amount (Michel Bruneau et al. 1998).

Before Northridge earthquake most of the beam-column connections in a moment

resisting frames were detailed to be able to transfer plastic moment of the beams to the

columns (Roeder and Foutch 1995). Thus, relatively lighter column and beam sizes were

sufficient for those frames to resist seismic forces. With time many engineers concluded

that it was economically advantageous to limit the number of bays in a frame designed as

ductile moment frame. Prior to the Northridge earthquake even some engineers

frequently designed building with only four single-bay ductile moment frames with two

in each principal direction. This trend results with the loss in structural redundancy. In an

addition these single-bay moment frames required considerably deeper beams and

columns with thicker flanges than the multi-bay ones previously used to resist the same

14

lateral forces. It provided an opportunity to investigate potential size effects (Michel

Bruneau et al. 1998). Roeder and Foutch (1996) compiled all the test results for pre-

northridge connections and found that the expected ductility decreases with the deeper

sections. Bonowitz (1999a) found the same results from the tests conducted after

Northridge earthquake (FEMA-355e)

POST-NORTHRIDGE DESIGN

Numerous factors have been identified which were potentially contributed to the

poor seismic performance of the pre-Northridge steel moment connections. Failure

happened due to different combinations of those factors: workmanship and inspection

quality; weld design; fracture mechanics; base metal elevated yield stress; welds stress

condition; stress concentrations; effect of triaxial stress conditions; loading rate; amd

presence of composite floor slab (Michel Bruneau et al. 1998).

Numerous solutions to the moment frame connection problems have been

proposed. Two key strategies have been developed to overcome the problem. One of

them is to strengthening the connection and the other is weakening the beam ends which

is framed into the connection. Both strategies effectively move the plastic hinges away

from the face of the column (Michel Bruneau et al. 1998).

Satisfactory performance requires that a connection should be capable of

developing a beam plastic rotation of 0.03 radian with a minimum strength equal to 80

percent of the plastic strength of the girder. These are the acceptance criteria suggested in

15

SAC interim guidelines (1995b). As the minimum requirement it was recommended that

experimental validation of proposed connection be done with the qualification tests in

compliance with the ATC-24 loading protocol (ATC1992).

SEMI-RIGID CONNECTIONS

A connection in a moment frame will termed as partially restrained if it

contributes to a minimum 10% of the lateral deflection or the connections strength is less

than the weaker element of connected members (FEMA-356). It is presumed that in the

Northridge earthquake, partially restrained connections could result in a better

performance to provide flexibility in the structure. Proper placing of semi-rigid

connections along with the rigid connection could improve the performance of moment

frames. Modeling process of semi-rigid connections in a beam column joint is described

in Chapter 4.

Kasai et al. (1999) and Maison et al. (2000) studied the effect of semi-rigid

connections within the SAC program. But, in those studies all the connections were

considered as partially restrained (FEMA-355c). However, the knowledge about the

effect of semi-rigid connections in a hybrid frame is limited. Built on the pioneer work of

Radulova (2009), this research aims to study the seismic performance of fully rigid and

hybrid rhombus and rectangular framing systems.

16

1.4 Objective of the Present Study

The objective of this study is to identify the effect of building shape on its

seismic performance. Thus rhombus shape frames are selected based on engineering

intuition and they were subjected to five earthquake records from four different frequency

earthquake. The control frame was selected to be the rectangular frame based on the SAC

Frame (FEMA-355) geometry. The aspect ratio and geometrical dimensions were varied

to obtain their earthquake response which included lateral sway, inter-story drift and

member forces. In the second phase of the study, high rise hybrid frames with rigid and

semi-rigid connections are subjected to earthquake forces to identify the effect of semi-

rigid connections on the global behavior of the frames.

1.5 Scope of the Study

The scope of this study was limited between Rhombus and Rectangular shape

frames. For both rhombus and rectangular; three, nine, twelve and twenty story frames

are modeled with all the connections being rigid. The frames are analyzed under 5

different seismic excitation records only. These are for Elcentro, Northridge, Parkfield

and two records for Kocaeli earthquakes. The results compared between the two shapes

includes lateral sway, inter story drifts and internal forces of the members. Semi-rigid

connection property has been incorporated only on the 20-story frames. Bi-linear

moment-rotation curve has been used to define semi-rigid connection properties. No

ultimate moment provision has been considered in moment-rotation curves. To create the

17

hybrid frames semi-rigid connections are placed only on the mid levels of the frames,

based on Radulova (2009).

1.7 Organization of the Present Study

This study is organized as per objectives. Background about this research is

narrated in Chapter 1. It includes previous earthquakes effects, seismic regulations, brief

discussion on connections, different framing systems and objectives of the study. Chapter

2 describes about the different analysis procedures. In Chapter 3, the proposed rhombus

frames are compared with the corresponding rectangular frames under different

earthquake records. Modeling process of semi-rigid connection is described in Chapter 4.

It also includes earthquake response of hybrid frames with rigid and semi-rigid

connections. Comments and discussions on the results are described in Chapter 5 along

with the recommendations for future work in this area.

18

CHAPTER 2

BACKGROUND ON FRAME ANALYSIS

2.1 Introduction

Structural analysis uses the laws of physics and mathematics to predict the

performance and behavior of structures. Actual behavior of a structure is complicated, but

different level of idealization could decrease the complexity. Here in this chapter the term

analysis will mainly deal with the procedures and guidelines to obtain the member

strength and deformation demands of a structure under seismic load. In the first part of

this chapter different analysis methods proposed by Federal Emergency Management

Agency (FEMA), National Earthquake Hazards Reduction Program (NEHRP) will be

overviewed. Then general discussion on some analysis procedure will be given briefly

and at last some solution techniques will be discussed.

2.2 Analysis Methods

Different degrees of complexity due to geometry of the structure along with the

material behavior are associated in the structural response under seismic excitation.

Depends on the needed accuracy different types of idealizations are proposed in the

evaluation of structural response. FEMA-355f considered four elastic and three inelastic

analysis procedures implemented by FEMA-273, NEHRP Guideline (1997) for

performance evaluation of steel moment resisting frames.

19

2.2.1 Elastic Analysis Methods

The proposed elastic analysis procedures are equivalent lateral force and modal

analysis by FEMA-302, FEMA-273 linear static and linear dynamic methods and linear

time history analysis procedures (FEMA-355f).

In the equivalent lateral force method, base shear is calculated based on seismic

response coefficient and total dead load with applicable portion of other loads. This base

shear is distributed to different floor levels and the response will be calculated from the

static analysis (FEMA-355f).

FEMA-273 linear static procedure uses the same background as the equivalent

lateral load method to calculate the seismic load. Instead of design base shear this method

brings in the term “pseudo lateral load” which is the final product after using different

modification factors. The “pseudo lateral load” is selected in a way so that the response

due to this load will be approximately same as from nonlinear time history analysis

(FEMA-355f).

Linear time history procedure uses two methods in calculating structural response.

In one method modal analysis and mode superposition is used which is explained later in

this chapter. In another method this procedure uses direct integration technique in

calculating seismic response. Some common integration techniques namely Newmark

and Wilson methods are discussed at the end of this chapter.

20

2.2.2 Inelastic Analysis Methods

The considered inelastic analysis procedures are FEMA-273 nonlinear static

procedure, capacity spectrum procedure (Skokan and Hart, 1999) and nonlinear time

history analysis (FEMA-355f).

FEMA-273 nonlinear static procedure is also known as the static pushover

analysis. In this method inelastic material behavior with P-∆ effects are included. These

effects are briefly discussed later. In this method a target displacement is assigned at any

point of the structure and then the structure is pushed with a increasing lateral load until

the target displacement is achieved to that point or the structure collapses (FEMA-355f).

Capacity spectrum method is mainly applicable for reinforced concrete structures

(ATC, 1996) and thus beyond the scope of this study. And the nonlinear time history

analysis procedure is same as the linear time history analysis, only material nonlinearity

along with the geometric effects should be considered in evaluation of structural

response. Some key terms of these analysis procedures are explained in the following

sections.

2.3 First Order Elastic Analysis

The first order elastic frame analysis considers the structural behavior as linear

under any type of loading. This analysis method does not consider the geometric effects

as member and the structural deflections (P-Δ and P-δ effects) as well as material

21

nonlinearity. It assumes that the displacement will be very small and therefore the second

order effects due to geometrical changes are ignored. And thus the stiffness matrix is

constant for the members independent of applied axial forces. The deflection is

proportional with the applied load, ie with increasing load the displacement will also

increase which can be expressed as a straight line as shown in Figure 2.1.

Figure 2.1 Generalised Load-Displacement curve for different types of analysis

The initial slopes of other types of analyses are coincided with it. It is because at

lower loads the structures do not develop any secondary effects in geometry and material

properties.

2.4 Second Order Elastic Analysis

Second order elastic analysis considers the geometric effects which are due to the

member and structural deflections named as P-δ and P-Δ effects respectively. Structural

response due to the second order elastic analysis is presented in the load-displacement

22

curve shown in Figure 2.1. Initially it follows the path of linear analysis, but as the

loading increased to produce sufficient geometric effect it starts to deviate from linear

analysis to show the effect of geometric nonlinearity. Figure 2.2 shows P-δ and P-Δ

effects, which is the cause for this geometric nonlinearity.

Figure 2.2 P-δ and P-Δ effects

These geometric effects produce higher internal forces due to axial loads. The

stiffness matrix needs to be adjusted to reflect these effects and the corrections develop

additional deflection. To account this problem the system reaches in equilibrium in an

iterative process. As the stiffness matrix and thus the structural response are dependent on

the deflected shape of the frame, principle of superposition is not applicable in second

order elastic analysis. As material nonlinearity is not considered, this method over

predicts the collapse load.

P P

δ

Δ

23

P-δ EFFECT

When a member deforms, it affects the stiffness of that member and additional

moment will be developed in the member. This second order effect due to deflection

along a member and the axial force is termed as P-δ effect.

P-Δ EFFECT

If a structure deflects significantly, then the original geometry of the structure

cannot be used to formulate the transformation matrix due to change in nodal coordinates.

This is known as P-Δ effect.

2.5 Inelastic Analysis

Material yielding and instability of structural members have significant effect to

control the ultimate load. If the material nonlinearity is included in the second order

elastic analysis, it would become the inelastic analysis. Nonlinearity in the inelastic

analysis exists in two forms – geometric nonlinearity and material nonlinearity.

Material start yielding at the outer fiber of the section when the elastic moment

reaches to yield moment value My. Material nonlinearity come into affect at this point

and after that with application of additional load, yielding will spread over the section

from outer fiber to plastic neutral axis. The section will continue yielding until the whole

section is yielded to develop full plastic moment Mp. This nonlinearity can be

incorporated to the analysis mainly by two approaches. In the first approach plastic

hinges are assumed to form at the two ends of a member i.e. all the material nonlinearity

24

is basically lumped at the two ends of a member. This is known as concentrated plasticity

(plastic hinge, lumped plasticity) approach. Another approach assumes the plasticity over

the whole member known as distributed plasticity (plastic zone) approach (Chan and

Chui, 2000).

CONCENTRATED PLASTICITY APPROACH

Concentrated plasticity approach ignores the progressive yielding along the

member length. It assumes the material nonlinearity to be lumped in a small region of

zero length (Yau and Chan, 1994). It deals with the cross section plastification, which

starts at the outer most fiber of the section and ends up with the formation of a hinge at a

given point. It assumes that the hinges will form only at the ends and the remaining part

of the member will remain elastic. Mashary and Chen (1991) and Yau and Chan (1994)

modeled this material nonlinearity by using zero length spring at the member ends. Many

other methods are proposed for computer simulation of this hinge property. This

approach is easier to apply and significantly save computation time than the distributed

plasticity approach. Commonly used methods for modeling plastic hinges are elastic-

plastic hinge method, column tangent modulus method, beam-column stiffness

degradation method, beam-column strength degradation method and end spring method.

DISTRIBUTED PLASTICITY APPROACH

This approach assumes yielding will be distributed over the length of the member

and the cross section. This method discretized structure into many elements and each

section is further divided into small fibers in an attempt to monitor stress and strain for all

members. Initial imperfections and residual stresses can be included by assigning stresses

25

to each fiber before loading which can be varied along the side and thickness of the

section (Chan, 1990). The distributed plasticity approach is more accurate than the

concentrated plasticity approach as fundamental stress-strain relationship is directly

applied for the computation of forces. This approach requires considerable amount of

computation time and huge memory to store data. Thus this method is suitable to analyze

simple structures. Commonly used methods for this approach are traditional plastic zone

method and simplified plastic zone method.

2.6 Dynamic Analysis of frame

Structural dynamics deal with the behavior of structure under dynamic loading. A

static load is one which does not vary with time. A dynamic is any load which changes its

magnitude, direction or position with time. If it changes very slowly, the structures

response may be determined using static analysis but if the loading varies with time

quickly (corresponding to the structures time period), the structures response must be

determined using dynamic analysis. Here in this study dynamic term will use for seismic

loads.

A structural dynamic analysis differs from the static analysis in two ways. Firstly

in a dynamic problem both the applied force and the resulting response in the structure

are time variant, i.e. function of time. It does not have a single solution like the static

problem. For complete evaluation of structural response one has to investigate the

solution over a specific interval of time. And the other one which is the most important

26

feature in dynamic analysis is the act of inertia force. If a load is applied in a structure

dynamically, there will be time variant deflection in the structure which will cause

acceleration and thus inertia force will be induced. Magnitude of the inertia force depends

on the acceleration and mass characteristics of the structure. Unlike static analysis,

dynamic problems significantly depend on mass and damping. Three components, mass

and damping together with the stiffness characteristics are required to write the equations

of motion. Mass will be calculated from all the loads that the structure carries and

members self-weight. This mass can be lumped at the joints or distributed over the

member. Stiffness is basically provided by the structural components of a system. And

the third component Damping is the energy dissipation properties of a material or system.

It is a process by which a free vibration could steadily diminishes in amplitude and

finally come to rest. In a vibrating system the energy could be dissipated by various

mechanisms and often more than one mechanism can be present at the same time.

The equivalent lateral load approach discussed before transforms dynamic force

into static forces. But, it cannot reflect the true dynamic response, as characteristic of

resonance cannot be explained in a static approach. To include all the dynamic effects in

the analysis, mode superposition and modal analysis is a widely accepted method for

linear systems. Different types of direct integration methods are used to solve both linear

and nonlinear dynamic problem numerically. Different methods are briefly discussed in

the following sections.

27

2.7 Modal Analysis

Modal analysis in structural dynamics is used to determine the natural mode

shapes and frequencies of a structure. It is a convenient method for calculating the

dynamic response of a linear structural system. The response of a MDF system under

externally applied dynamic load can be described by N differential equations in the

following form,

[m] {ü} + [c]{ů} + [k]{u} = {p(t)}, where

[m] is the mass matrix,

[c] is the damping matrix,

[k] is the stiffness matrix of the system,

{p(t)} is the externally applied dynamic force matrix, and

{u}, {ů}, {ü} denotes displacement, velocity and acceleration matrix.

The main approach of this method for dynamic analysis is to change N set of

coupled equations of motion into N uncoupled equation for a multiple-degree-of-freedom

system. Based on the number of DOF, a MDF system has multiple characteristic

deflected shapes. Each characteristic deflected shape is called a natural mode of vibration

of the MDF system denoted by øn. The displacement {u(t)} of the system can be

determined by the superposition of modal contributions. i.e. {u(t)} = ∑=

N

n 1

qn(t)øn, where

qn(t) = modal coordinates. Deflected shape øn does not vary with time. The equation,

[k]øn = ωn2[m]øn , is the matrix eigen value problem where ωn is the natural frequency and

28

øn is the natural modes of vibration of the system (Chopra, A.K. 1995). This equation has

a non trivial solution if,

0][][ 2 =− mk nω ,

This is known as the frequency equation, because after expanding the determinant

it creates a polynomial of order N in ωn2. This equation has N number of real and positive

roots for ωn2 i.e. N number of natural vibration frequencies, started with ω1 be the

smallest and ωn be the largest. If applied force {p(t)} can be written as [s]p(t) with spatial

distribution is defined by [s], then the spatial distribution is expanded to its modal

components {sn}. where, {sn} = Γn [m]øn. Then the equation could be transformed to

uncoupled equations in modal coordinates and the solution for the modal coordinate is,

qn(t)= ΓnDn(t), where Dn is governed by the equation of motion for nth-mode SDF system

of the nth mode of the MDF system. The contribution from this mode to modal

displacement is, {un(t)} = ønqn(t) = ΓnønDn(t). And the equivalent static force associated

with the nth mode response is, {fn(t)}={sn}{An(t)}, where An(t) = ωn2Dn(t), is the pseudo

acceleration. The nth mode contribution to any response is determined by the static

analysis for force {fn}. By combining all the response contributions from all the modes

gives the total dynamic response (Chopra, A.K. 2007).

2.8 Step-by-Step Integration

Analytical solution of a dynamic problem is not possible in cases where the

physical properties like geometry and elasticity of material do not remain constant. The

stiffness coefficient can be changed with yielding of materials or by significant change in

29

axial force which will cause the change in geometric stiffness coefficient. The applicable

method for the analysis of nonlinear system is the numerical step-by-step integration,

which is also applicable to linear systems. The main approach of this method is to divide

the response history into short time increments and then the response is calculated during

each increment assuming it as a linear system with the properties determined at the start

of each increment. The properties are updated at the end of each interval based on the

deformation and stress, and thus the nonlinear system is idealized as a collection of

changing linear systems. In 1959, Newmark N.M. developed a method of computation

for dynamic problems in structure. Later modifications are made to this method based on

the stability, accuracy etc. Some popular methods are briefly discussed in the following

sections.

2.8.1 Newmark’s Method

In this method it is assumed that at time i the values of displacement, velocity and

acceleration is known and by numerical integration it can be estimated for time i+1, if the

time increment, ∆t is very small. Newmark introduce two parameters γ and β to indicate

the proportion of acceleration that will enter into the equations for displacement and

velocity (Newmark N.M. 1959). The adopted equations are,

ui+1 = ui + (∆t) ůi + [ (0.5 – β)(∆t)2] üi + [ β(∆t)2 ] üi+1 (2.1)

ůi+1 = ůi + [ (1 – γ) ∆t ] üi + (γ ∆t) üi+1 (2.2)

The two parameters γ and β in the above two equations are responsible for the

stability and accuracy of the system. If γ is taken as zero, a negative damping will result,

which will produce a auto vibration from the numerical method. Again if γ is greater

30

than21 , a positive damping addition to the real damping will be introduced to reduce the

magnitude of the response. So, generally γ is taken as equal to 21 . And better results are

obtained with values of β range between 61 to

41 (Newmark, N.M. 1959). Two popular

special cases for Newmark’s method are as follows -

2.8.1.1 Average Acceleration Method

Figure 2.3 Average acceleration

If there is no variation of acceleration over a time step and the value is constant,

equal to the value of average acceleration, then,

ü ( )τ = 21 (üi+1+ üi) . If ( )τ = ∆t , this will yield

ůi+1 = ůi + 21 ∆t (üi + üi+1) , and ui+1 = ui + (∆t) ůi +

41 (∆t)2 (üi + üi+1).

τ

Δt

ü

t ti ti+1

üi+1

üi

31

These two equations are same as equation 2.1 and 2.2, if γ =21 and β =

41 . For

this method, maximum velocity response is correct whether value of β other than 41 will

cause some error (Newmark N.M. 1959). From stability point of view Newmark’s

method is stable if,

βγπ 2

12

1−

≤Δ

nTt (2.3)

Where, Tn is the natural time period of the system.

For, γ =21 and β =

41 , ∞≤

Δ

nTt ie average acceleration method is unconditionally stable.

2.8.1.2 Linear Acceleration Method

Figure 2.4 Linear acceleration

If the variation of acceleration is linear over a time step, then,

ü ( )τ = üi + tΔτ (üi+1 – üi). If ( )τ = ∆t , this will yield

τ

Δt

ü

t ti ti+1

üi+1

üi

32

ůi+1 = ůi + 21 ∆t (üi + üi+1) , and

ui+1 = ui + (∆t) ůi + (∆t)2 (31 üi +

61 üi+1).

These two equations are same from equation 2.1 and 2.2, if γ =21 and β =

61 .

Equation 2.3 shows that the linear acceleration method will be stable, if 551.0≤Δ

nTt . And

so this method is conditionally stable. But stability criteria have not imposed any rule in

selecting time step. In general by taking short integration interval, a good accuracy can be

achieved from unconditionally stable linear acceleration method.

2.8.2 Wilson θ Method

E.L.Wilson modified the conditionally stable linear acceleration method into

unconditionally stable. His proposed method is known as Wilson θ Method. This method

assumes that the acceleration will vary linearly over an extended interval, δt = θ∆t.

Figure 2.5 Linear variation of acceleration over extended time steps

t Δt

ü

ti ti+1

üi+1

üi

ti+θ

Δüi δüi

δt= θ Δt

33

The parameter θ in this method determines the accuracy and the stability

characteristics of the numerical analysis. For θ = 1, this method will turn into Newmark’s

standard linear-acceleration method. For θ ≥ 1.37, Wilson’s method becomes

unconditionally stable.

2.8.3 Hilber-Hughes-Taylor Method

In order to introduce numerical damping into Newmark’s method without

affecting the accuracy Hilber, Hughes and Taylor introduce the parameter α. Where,

γ = ( )221 α− and β = ( )

41 2α−

The range of this parameter is from - 31 to 0. If α = 0, then this method turns into

Newmark’s average acceleration method. This would give the higher accuracy but it may

produce excess vibrations in the higher modes. Decreasing value of α will increase the

amount of numerical damping which will mainly damp the higher frequency modes.

Sometimes it is required for a nonlinear solution to converge with using negative α. value

(SAP2000).

34

CHAPTER 3

EFFECT OF FRAME SHAPE AND GEOMETRY ON SEISMIC PERFORMANCE

3.1 Introduction

The SAC joint venture between the Structural Engineers Association of California

(SEAOC), the Applied Technology Council (ATC), and the California Universities for

Research in Earthquake Engineering(CUREe), proposed 3-story, 9-story and 20-story

frames. Several steel frames with these SAC dimensions are used for this study. These

frames represent from low-rise to high-rise buildings. Besides these, a 12-story frame is

also modeled. Initially all buildings are modeled as Rectangular Moment Frame with all

connections being rigid. Later, a Rhombus shape has been proposed to see the effect of

shape on the global behavior of the frame. The Rhombus shape has been constructed by

placing the Rhombus building inside the Rectangular frame. This Rhombus shape frame

has the same height and width ratios as the Rectangular shape. All the member sizes,

joint loads and uniformly distributed loading were kept the same for both the shapes.

All the frames are analyzed under five different earthquake excitation data. These

earthquakes reflect a wide variation in frequency content. For each height, results are

compared between Rectangular and Rhombus shapes. Results include lateral sway, drift,

35

top displacement and member forces. Effect of height-to-width ratios on the

fundamental properties of the structure is also observed.

Total eight rigid frames are developed. To describe the results a general building

designation is adopted for convenience,

_S – H_W_H/W- OP( _,/ _ ) - EQ , where

S = story

H = the height of the frame

W = the width of the frame

H/W = the height-to-width ratio of the frame

OP = out put, which can be member forces or displacement

AF = axial force

SF = shear force

BM = bending moment

TDUx = top node displacement along X-direction

MDUx = maximum displacement along X-direction

LDr = Lateral inter story drift

RMR = Rigid Rhombus frame

RCR = Rigid Rectangular frame

EQ = earthquake

ELC = El Centro earthquake

NRG = Northridge earthquake

36

PKF = Parkfield earthquake

KCL1 = Afyon Bay record for Kocaeli earthquake

KCL2 = Aydin record for Kocaeli earthquake

The building designation follows: 20S – 289_100_2.89 – TDUx (RMR, RCR) –

ELC, which will describe a 20 story frame with height-to-width ratio of 2.89 where the

height and the width are 289 ft and 200 ft, respectively. The out put describes top

displacement along X- direction for both the Rhombus shape and Rectangular shape with

all the connections being rigid, and the results are for El Centro earthquake. Thus, BM(

RMR/RCR) will describe bending moment as out put. And the result will be the ratio of

bending moment for the Rhombus shape and the Rectangular shape.

3.2 Three Story Frame

The analysis starts with the modeling of a three story frame. This three story 4-

bay frame is used to observe the effect of earthquake excitation on low rise structures.

The rectangular frame is 120ft wide with 30ft bay dimension and the total height of the

building is 39 ft. These dimensions are taken from SAC 3-Story building. All the member

sizes are W14 X 283. All the connections are considered to be rigid. Total dead load and

25 percentage of live load are considered for calculating member mass. Load calculations

are described in Appendix G. The 3-story rectangular frame with corresponding rhombus

shape is shown in Figure 3.1. These frames are analyzed under five different earthquake

excitation records. The results are plotted in the Appendices A through F of this report as

an attempt to show the comparison between the two shapes.

37

Figure 3.1 Three-Story Rectangular and Rhombus frame

Both of these models are analyzed under modal analysis and the corresponding

first couples of mode shapes are given below.

(a) (b)

Figure 3.2 First two Mode Shapes for 3-Story Rectangular Frame (a) Mode 1, (b) Mode 2

(a) (b)

Figure 3.3 First two Mode Shapes for 3-Story Rhombus Frame (a) Mode 1, (b) Mode 2

These mode shapes are obtained from the SAP2000 software. Corresponding

natural time period and frequencies for first ten modes are tabulated in the following

table.

38

Table 3.1 Modal period and frequencies for 3-Story Frames (SAP 2000)

Mode 3S – 39_120_0.325 – (RCR) 3S – 39_120_0.325 – (RMR)

Period Frequency Period Frequency (sec) (Hz) (sec) (Hz)

1 0.56324 1.7754 0.28961 3.4529 2 0.17082 5.8543 0.17561 5.6943 3 0.0964 10.374 0.11076 9.0288 4 0.05435 18.4 0.06529 15.317 5 0.05406 18.499 0.0555 18.019 6 0.05368 18.63 0.05294 18.888 7 0.0479 20.877 0.04959 20.167 8 0.04613 21.679 0.03818 26.191 9 0.04231 23.637 0.03408 29.344

10 0.03836 26.068 0.02795 35.777

3.3 Nine Story Frame

The nine story frame has been modeled with the SAC 9-Story building

dimensions. This nine story 5-bay frame is used to observe the effect of earthquake on

medium high-rise buildings. The rectangular frame is 150ft wide with 30ft bay

dimension, and the total height of the building is 134 ft with a 12ft basement. All the

member sizes are W14 X 283. All the connections are considered to be rigid. Total dead

load and 25 percentage of live load are considered for calculating member mass. Load

calculations are described on Appendix G. The 9-story rectangular frame with equivalent

rhombus shape is shown in Figure 3.4. These frames are analyzed under five different

earthquake excitation records. The results include top displacements, lateral sway, inter

story drifts and member forces. All the results are plotted in the Appendices A through F

39

of this report to compare the performance of individual shape under earthquake

excitations.

Figure 3.4 Nine-Story Rectangular and Rhombus frame

These models are analyzed under modal analysis and the first couples of mode

shapes are given in the following Figures.

(a) (b)


40

(a) (b)


The modal analysis is performed in the SAP2000 platform. Corresponding time

period and frequencies for first ten modes are tabulated in the following Table


Mode 9S – 134_150_0.893 – (RCR) 9S – 134_150_0.893 – (RMR)


1 1.92246 0.52017 0.4332 2.3084 2 0.6213 1.6095 0.37677 2.6542 3 0.3527 2.8353 0.18268 5.474 4 0.23647 4.2289 0.17649 5.6662 5 0.17226 5.8051 0.13888 7.2004 6 0.16484 6.0667 0.12993 7.6968 7 0.16 6.25 0.12041 8.3047 8 0.15317 6.5286 0.1063 9.4078 9 0.1468 6.8121 0.08393 11.915

10 0.13315 7.5105 0.07483 13.364

41

3.4 Twelve Story Frame

The twelve story 5-bay frame is used to observe the geometric effect of frame

under earthquake excitation. The rectangular frame is 150ft wide with 30ft bay

dimension, and the total height of the building is 173 ft with a 12ft basement. All the

member sizes are W14 X 283. All the connections are considered to be rigid. Total dead

load and 25 percentage of live load are considered for calculating member mass. The 12-

story rectangular frame with corresponding rhombus shape is shown in Figure 3.7. These

frames are analyzed under five different earthquake excitation records. The out put

results include top displacement, lateral sway, inter story drift and member forces. All the

results are plotted in the Appendices A through F of this report as an attempt to show the

comparison between the two shapes.

Figure 3.7 Twelve-Story Rectangular and Rhombus frame

42

Both of these frames are analyzed under modal analysis. In the following Figures,

first couples of mode shapes are provided. These shapes are obtained from the SAP2000

software.

(a) (b)


(a) (b)


43

Corresponding time period and frequencies for the first ten modes are tabulated

below.


Mode 12S – 173_150_1.153 – (RCR) 12S – 173_150_1.153 – (RMR)


1 2.51114 0.39823 0.5671 1.7634 2 0.82114 1.2178 0.53406 1.8725 3 0.47433 2.1082 0.25183 3.9709 4 0.32474 3.0794 0.22645 4.4159 5 0.24064 4.1556 0.19174 5.2154 6 0.20952 4.7728 0.16486 6.0658 7 0.20019 4.9952 0.1559 6.4144 8 0.18754 5.3323 0.1157 8.643 9 0.18752 5.3327 0.10739 9.3116

10 0.176 5.6818 0.09298 10.755

3.5 Twenty Story Frame

A 20story rectangular frame has been modeled with SAC 20-Story building

dimensions. This 20-Story 5 bay frame is used to observe the effect of earthquake on

high-rise buildings. The rectangular frame is 100ft wide with 20ft bay width and the total

height of the building is 289ft with two basement floor 12ft each. All the member sizes

are W24 X 131. All the connections are considered to be rigid. Total dead load and 25

percentage of live load are considered for effective seismic load. 20-story rectangular

frames with corresponding rhombus shapes are shown in Figure 3.10. Both of the frames

are analyzed with Modal analysis on the SAP2000 platform. First couples of modal

44

shapes obtained from modal analysis are given in Figure 3.11 and 3.12. Corresponding

time period and frequencies for first ten modes are given in Table 3.4. These frames are

analyzed under five different earthquake excitation records. The results include top

displacements, lateral sway, inter story drift and member forces. All the results are

plotted in the Appendices A through F of this report to compare the performance of

individual shape under earthquake excitation.

Figure 3.10 Twenty-Story Rectangular and Rhombus frames

45

(a) (b)


(a) (b)


46


Mode 20S–289_100_2.89–(RCR) 20S–289_100_2.89-(RMR)


1 2.72965 0.36635 1.58828 0.62961 2 0.87486 1.143 0.48492 2.0622 3 0.49 2.0408 0.279 3.5842 4 0.34352 2.9111 0.25668 3.8959 5 0.31924 3.1324 0.22956 4.3562 6 0.2623 3.8125 0.18013 5.5514 7 0.24444 4.0909 0.15376 6.5037 8 0.20718 4.8266 0.13834 7.2287 9 0.17824 5.6103 0.11651 8.5827

10 0.17207 5.8116 0.10433 9.5849

3.6 Eathquake in consideration

Ground acceleration values in an earthquake vary with time in an irregular

manner. As a result, an earthquake is generally composed of infinite number of frequency

content. The non periodic acceleration time function can be represented by the Fourier

integral –

ωωπ

ω deFtu tig ∫

∞

∞−

−= )(21)(&&

A Fourier spectrum constitutes the representation of a time history into the

frequency domain. If üg(t) denotes ground acceleration in the time domain, the Fourier

spectrum of üg(t) is defined as

dtetuF tig∫

∞

∞−

−= ωω )()( &&

47

F(ω) represents ground acceleration in frequency domain. The amplitude of the

Fourier spectrum is used to identify the harmonic components of the earthquake that has

the largest amplitudes. These harmonic components are identified in terms of their

frequencies. The amplitude Fourier spectrum of an earthquake may be interpreted as a

measure of the total energy contained to that ground motion.

For this study four different earthquakes are used with a wide variety of frequency

range. El Centro and Northridge earthquakes represent the high-frequency range,

Parkfield earthquake is a medium-frequency earthquake. And two site records are

provided for low frequency Kocaeli earthquake.

3.6.1 Generated Earthquake

To observe the resonance effect on 20 story frames, a Sine wave, composed of

three different time period is generated; ie it consists of total three frequencies. Another

reason for constructing this imaginary earthquake is to verify the SeismoSignal software

about it’s capability to determine dominant frequencies, as the original frequency

contents for this generated data are known. Total time for the generated earthquake is

79.17 sec. It is constructed in a manner, so that in the first part, it will create resonance

with the 20 story rhombus frame, and the last part will create resonance with the 20 story

rectangular frame. From 0 ~ 10.92 sec, time period, T = 1.365 sec; frequency, f

=0.733Hz, with total 8 cycle. In this time range amplitude is almost three times higher

than the rest of the vibration. From 10.92 ~ 54.6 sec, time period, T = 5.46 sec;

frequency, f = 0.183Hz, with total 8 cycle. And from 54.6 ~ 79.17 sec, time period, T =

48

2.73 sec; frequency f = 0.366Hz, with total 9 cycle. The generated data is plotted in

Figure 3.13.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 16 32 48 64

Time (sec)

Acc

eler

atio

n (g

)

Figure 3.13 Acceleration Time History of Generated Earthquake

Now, if this data is used in SeismoSignal software, it will give three dominant

frequencies as in Figure 3.14.

1st dominant, f = 0.183Hz,

2nd dominant, f = 0.733Hz, and

3rd dominant, f = 0.366Hz.

49

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5Frequency (Hz)

Four

ier A

plitu

de

Figure 3.14 Fourier Spectrum of Generated Earthquake

In Figure 3.14, 0.183Hz is the 1st dominant frequency, because it covers most of

the time range in Figure 3.13. But the software does not consider only the time range.

That is why 0.733Hz is the second dominant instead of 0.366Hz (which is the second

large in terms of time range). 0.733Hz is the 2nd dominant, as the amplitude of

acceleration of that part is almost three times higher than the other.

To observe the resonance effect, 20S–289_100_2.89–(RMR) and 20S–

289_100_2.89–(RCR) frames are analyzed with this data. The resulted top displacements

of the two frames are shown in Figure 3.15. Figure shows, rhombus frame gives top

displacement in the first part of the total time range, as vibration frequency of that time

range is close to the natural frequency of the rhombus frame. And the rectangular frame

50

gives maximum top displacement at the last part of the total time, where vibration

frequency is close to the natural frequency of the rectangular frame.

-120

-104

-88

-72

-56

-40

-24

-8

8

24

40

56

72

88

104

120

0 16 32 48 64 80Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-304.8

-244.8

-184.8

-124.8

-64.8

-4.8

55.2

115.2

175.2

235.2

295.2

Dis

plac

emen

t-Ux(

cm)

20S-289_100_2.89-TDUx(RMR)-genEQ 20S-289_100_2.89-TDUx(RCR)-genEQ

Figure 3.15 20S-289_100_2.89-TDux(RMR,RCR)-generated data

None of the frames experience higher top node displacement at the middle part of

the generated data, which one is the dominant frequency range for the total time. Again,

the rectangular frame which has resonance effect with the third dominant frequency,

gives a maximum top displacement value, which is more than double, than the top

displacement for the rhombus frame. So, in the following study, frequencies of the

earthquakes are defined as a range instead of giving emphasis on a single dominant

frequency.

51

3.6.2 ElCentro

The epicenter of the El Centro earthquake was in El Centro (Imperial Valley),

California. It occurred on May 18, 1940. The exact site, from where data was collected is

Imperial Valley, which is located 6 miles away from the epicenter. Nine people were

killed by the earthquake. At Imperial, 80 percent of the buildings were damaged to some

degree. In the business district of Brawley, all structures were damaged, and about 50

percent had to be condemned. The shock caused 40 miles of surface faulting on the

Imperial Fault, part of the San Andreas system in southern California. Total damage has

been estimated at about $6 million. The instrument that recorded the accelerogram was

attached to the El Centro Terminal Substation Building’s concrete floor. The record may

have under-represented the high frequency motions of the ground because of soil-

structure interaction of the massive foundation with the surrounding soft soil. The

earthquake magnitude was 7.1. The total duration was 53.5 seconds. The North-South

component of the El Centro earthquake was recorded with 0.02 second interval. First 30

seconds of the earthquake are used for time history analysis. Elcentro represents a high

frequency ground shaking. Its major frequency range is between 1.0 Hz to 2.25 Hz based

on the Fourier spectrum obtained from the SeismoSignal software. Figure 3.16 represents

the acceleration time history of El Centro earthquake in terms of the gravitational

acceleration, g. Figure 3.17 represents the Fourier transformation of the earthquake.

52

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30Time (sec)

Acc

eler

atio

n (g

)

Figure 3.16 Acceleration Time History of El Centro Earthquake

0

0.05

0.1

0.15

0.2

0.25

0.3

0 5 10 15 20 25 30Frequency (Hz)

Four

ier A

plitu

de

Figure 3.17 Fourier Spectrum of El Centro Earthquake

53

Table 3.5 Frequency Content for El Centro Earthquake

Earthquake : Elcentro Major frequency range 1.0Hz ~ 2.25Hz

Dominant Frequency order Frequency (Hz)

Power Amplitude

Fourier Amplitude

1st 1.172 0.171 0.25 2nd 2.148 0.127 0.216 3rd 1.782 0.124 0.213 4th 2.075 0.119 0.209

3.6.3 Northridge

The epicenter of Northridge earthquake was in Northridge, California. The

Northridge earthquake was occurred at 4:30 a.m. local time on January 17, 1994. This

earthquake is significant, as the damage created by this earthquake put a question mark

on the design philosophies followed by the engineers at that time. The earthquake

occurred along a "blind" thrust fault, close to the San Andreas Fault. The data was

collected from Sylmar, Olive View. The earthquake magnitude is 6.7 on the Richter

scale. The number of fatalities in the Northridge earthquake was 57. About 9000 people

were injured. The total duration was 59.9 seconds, recorded with 0.02 second interval.

Northridge is a high frequency earthquake with its major frequency between 0.4 Hz to 3.5

Hz, based on the Fourier spectrum obtained from the SeismoSignal software. Figure 3.18

represents the acceleration time history of Northridge earthquake in terms of the

gravitational acceleration, g. Figure 3.19 represents the Fourier transformation of the

earthquake. The first 50 second of the record has been used for time history analysis.

54

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60Time (sec)

Acc

eler

atio

n (g

)

Figure 3.18 Acceleration Time History of Northridge Earthquake

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 3 6 9 12 15Frequency (Hz)

Four

ier A

plitu

de

Figure 3.19 Fourier Spectrum of Northridge Earthquake

55

Table 3.6 Frequency Content for Northridge Earthquake

Earthquake : Northridge Major frequency range 0.4Hz ~ 3.5Hz


Power Amplitude

Fourier Amplitude

1st 0.415 0.207 0.318 2nd 1.123,1.16 0.175 0.293 3rd 3.271 0.164 0.284 4th 1.587 0.138 0.26

3.6.4 Parkfield

The epicenter of Parkfield earthquake was in Parkfield, California. It occurred on

28th of June, 1966. The data was collected in station Cholame #2, 065 (CDMG STATION

1013). The earthquake magnitude was 6.1 on the Richter scale, and the total duration of

the record was 43.7 seconds. The ground acceleration data was taken with 0.01 sec

interval. Parkfield represents the medium-frequency earthquake. Its major frequency

range is between 0.5 Hz to 1.75 Hz based on the Fourier spectrum obtained from the

SeismoSignal software. Figure 3.20 represents the acceleration time history of Parkfield

earthquake in terms of the gravitational acceleration, g. Figure 3.21 represents the

Fourier transformation of the earthquake. The first 40 seconds of the record are used for

time history analysis.

56

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30 35 40

Time (sec)

Acc

eler

atio

n (g

)

Figure 3.20 Acceleration Time History of Parkfield Earthquake

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10 12 14Frequency (Hz)

Four

ier A

plitu

de

Figure 3.21 Fourier Spectrum of Parkfield Earthquake

57

Table 3.7 Frequency Content for Parkfield Earthquake

Earthquake : Parkfield Major frequency range 0.5Hz ~ 1.75Hz


Power Amplitude

Fourier Amplitude

1st 1.428 0.174 0.251 2nd 0.781 0.166 0.246 3rd 1.648 0.158 0.24 4th 0.708 0.142 0.227

3.6.5 Kocaeli

Kocaeli earthquake occurred on August 17, 1999 in the North Anatolian Fault

Zone with a macro seismic epicenter near the town of Gölcük, a sub-province of Kocaeli

in the western part of Turkey. Two earthquake data from two site locations are used for

this earthquake; one is Afyon Bay,N (ERD), Turkey and the other is Aydin,S (ERD),

Turkey. The earthquake magnitude was 7.4 on the Richter scale. The total duration of the

record for each site was 180.6 and 220.2 seconds respectively. For Afyon Bay N(ERD),

data were recorded with 0.0078125 second interval, and for Aydin S(ERD), data were

recorded with 0.01 second interval. Kocaeli can be represented as a low frequency

earthquake. Its major frequencies for Afyon Bay, N(ERD) record range between 0.80 Hz

to 1.2 Hz, and for Aydin, S(ERD) record, this range is between 0.25 Hz to 0.75 Hz. These

ranges are based on the Fourier spectrum obtained from the SeismoSignal software.

Figure 3.22 and 3.24 presents the acceleration time history of Kocaeli earthquake in

terms of the gravitational acceleration, g for two different records. Figure 3.23 and 3.25

represents the Fourier transformation of the corresponding earthquake records.

58

-0.015

-0.012

-0.009

-0.006

-0.003

0

0.003

0.006

0.009

0.012

0.015

0 45 90 135 180Time (sec)

Acce

lera

tion

(g)

Figure 3.22 Acceleration Time History of Kocaeli[Afyon Bay,N(ERD)] Earthquake

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 1 2 3 4 5 6Frequency (Hz)

Four

ier A

plitu

de

Figure 3.23 Fourier Spectrum of Kocaeli[Afyon Bay,N(ERD)] Earthquake

59

Table 3.8 Frequency Content for Kocaeli[Afyon Bay,N(ERD)] Earthquake

Earthquake : Kocaeli[Afyon Bay,N(ERD)] Major frequency range 0.80Hz ~ 1.2Hz


Power Amplitude

Fourier Amplitude

1st 0.98 1.496 0.059 2nd 0.898 0.705 0.04 3rd 0.922 0.611 0.038 4th 0.848 0.584 0.037

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 50 100 150 200Time (sec)

Acc

eler

atio

n (g

)

Figure 3.24 Acceleration Time History of Kocaeli[Aydin,S(ERD)] Earthquake

60

0

0.005

0.01

0.015

0.02

0.025

0.03

0 1 2 3 4 5 6Frequency (Hz)

Four

ier A

plitu

de

Figure 3.25 Fourier Spectrum of Kocaeli[Aydin,S(ERD)] Earthquake

Table 3.9 Frequency Content for Kocaeli[Aydin,S(ERD)] Earthquake

Earthquake : Kocaeli[Aydin,S(ERD)] Major frequency range 0.25Hz ~ 0.75Hz


Power Amplitude

Fourier Amplitude

1st 0.308 1.258 0.029 2nd 0.33 1.184 0.028 3rd 0.406 1.164 0.028 4th 0.299 1.044 0.026

61

Dominant frequency content for all the earthquakes considered are given below :

Table 3.10 Dominant frequency order for the considered earthquake

Dominant

Frequency

Order

Major Earthquake Frequency Range (Hz)

Elcentro Northridge Parkfield Kocaeli

Afyon Bay

Kocaeli

Aydin

1st 1.172 0.415 1.428 0.98 0.308

2nd 2.148 1.123,1.16 0.781 0.898 0.330

3rd 1.782 3.271 1.648 0.922 0.406

4th 2.075 1.587 0.708 0.848 0.299

3.7 Discussion of the Results

This study starts with the analysis of low rise to high rise steel frames, under

different earthquake excitations. Low rise structures are expected to be more vulnerable

in high frequency earthquakes than the high rise structures. On the other hand, high rise

buildings are more vulnerable in low frequency earthquakes comparing to the low rise

buildings. Rhombus shape frames are stiffer than the rectangular shape frames, thus the

natural frequency of rhombus frames are higher than the rectangular frames with same

aspect ratio. This is described in the previous sections from Table 3.1 to Table 3.4.

In the 3-story, 9-story, 12-story and 20-story frames, rhombus shapes experienced

less displacement than the rectangular frames in earthquake analysis. This effect is

discussed in terms of top node displacement, lateral sway profile of the frame, and inter

62

story lateral drift. The comparisons between rhombus and rectangular shapes in this

chapter are based on the assumption of all the connections being rigid.

Top displacement profiles for the whole time range for an earthquake are shown

in Figure A.1 through A.20. For all the frames with different earthquakes, rhombus shape

improves building performance than the rectangular shape. In most of the cases, top

displacement for rectangular frame is higher than the top displacement for the rhombus

frame. Only, for 20S-289_100_2.89 buildings, rhombus shape has comparable top node

displacement with the rectangular frame for Parkfield and Afyon Bay record of Kocaeli

earthquakes. This is due to the reason that, 20S-289_100_2.89 rhombus frame has natural

frequency close to the 4th dominant frequency for Parkfield and Afyon bay records.

Maximum lateral displacement of each floor is plotted from Figure E.1 through

E.20. These sway profiles show the maximum lateral displacement of each floor for the

total time range of an earthquake. Maximum lateral displacement may occur at different

time for different floor level. All the rhombus frames improve building performance than

the rectangular frame. But, for 20S-289_100_2.89 rhombus shape, some lower levels of

the frame experience a little bit higher displacement than the rectangular frames for

Parkfield and Afyon Bay records of Kocaeli earthquakes.

Lateral drift ratio for all the rigid frames are plotted in Figure F.1 through F.20.

These inter story drift ratios are calculated with the relative displacement of each floor

63

from the floor below, divided by the corresponding floor height. This parameter is very

significant in performance based engineering and should be in acceptable limit as per the

requirement of different performance levels. Most of the rhombus shape has less inter

story drift ratios compared with the rectangular shape. Only, in the case of 20S-

289_100_2.89 rhombus frame, few stories experience a little more drift than the

rectangular shape frame for Parkfield and Afyon Bay records for Kocaeli earthquakes.

Performance of the individual rigid frame shapes are closely observed also in

terms of internal forces. Axial force, shear force and bending moments are compared. For

comparison purpose, maximum internal force values from each floor for rhombus shape

is compared with the maximum value from each floor for rectangular shape. Both beams

and columns are compared. Diagonal elements for rhombus shapes are considered as

columns.

Most of the bending moment ratios between rhombus and rectangular frames in

Figure D.1 through D.20 are less than 1. For 3-story frames, bending moment for beams

and columns from rhombus frames are much lower than the rectangular frames. The

lowest bending moment demand in a column from rhombus frame is around 42 times

lower than the rectangular frame. And in case of beam, bending moment demand is

around 32 times lower than the demand for rectangular frame.

64

In the 9-story frames, all the bending moment ratios are less than one except in

one story, where demand increases around 17 percent in a column moment for El Centro

earthquake. The largest reduction for rhombus frame demand is 25 times lower in beam

moment and 33 times lower in column moment than the rectangular shape demand for

Aydin record of Kocaeli earthquake. For 12-story frames, most of the bending moment

demands for rhombus shape are less than the demands for rectangular frame. One story

requires around 30 percent increase in beam moment and two stories require maximum

24 percent increase in column moment for Parkfield earthquake. Aydin record of Kocaeli

earthquake reduces the demand for rhombus shape to maximum 454 times lower in the

beam moment and maximum 333 times lower in the column moment than the demand for

rectangular shape.

For 20-story rhombus frame, few bottom stories require higher moment than the

rectangular frame. The maximum demand for beam moment is 155 percent higher and

column moment is 482 percent higher than the rectangular frame for Parkfield

earthquake. Other than these few cases, in most of the stories, beam and column moment

demands are lower for rhombus frame than the rectangular frame. Rhombus shape

produces maximum 40 times reduction in beam moments and 53 times reduction in

column moments than the rectangular frame for Northridge earthquake.

Shear force ratios between rhombus and rectangular frame as in Figure C.1

through C.20 follow the same trend as bending moment ratios. All the shear force ratios

65

between 3-story rhombus and rectangular frames are less than 1. All the beam shear

forces in a 9-story rhombus frame are less than the rectangular frame. The largest

reduction is 25 times lower than the rectangular frame for Aydin record of Kocaeli

earthquake. One story demands 41 percent higher column shear force than the demand

for rectangular frame in ElCentro earthquake. Most of the elements in 12-story rhombus

frames have less shear force demand than the rectangular frame. The largest demand

reduction in shear for rhombus frame is 385 times lower for beam and 1310 times lower

for column than the rectangular frame when subjected to Aydin records of Kocaeli

earthquake. Demands increase in couple of stories for rhombus frame. Largest increase in

shear force demand for rhombus shape is around 2 times higher in beam shear and 24

percent higher in column shear force demand than the rectangular frame.

For 20-story frame, most of the shear force ratios are less than one. Only a few

bottom stories require higher shear force demand for rhombus frame. The largest shear

demand is 210 percent higher for beam and 133 percent higher for column than the

rectangular frame for Parkfield earthquake. The largest reduction in shear demand for

rhombus frame is 18 times lower for beam shear and 133 times lower for column shear,

than the rectangular frame for Northridge earthquake.

The members in a rhombus frame experience higher axial force than the

rectangular frames. This is due to the inherent property of the framing system, which uses

the axial stiffness to resist lateral loads. In most of the cases, axial force demands for

66

rhombus frames are higher than the demands for rectangular frames. In all the rhombus

shapes, beams in the mid levels experience higher axial loads. The largest demand for

beam axial force is 199 times higher in a 20-story rhombus frame than the rectangular

frame for Afyon Bay record for Kocaeli earthquake. And largest demand for column

axial force is 680 percent higher in a 12-story rhombus frame column than that of a

rectangular frame.

67

CHAPTER 4

HYBRID FRAME SYSTEMS

4.1 Introduction

High-rise 20-Story frames are considered in this study to observe the effect of

semi-rigid connections in resisting earthquake excitations. Rectangular frame with SAC-

20story dimensions and the equivalent rhombus frame have been used. Then, width of the

rhombus frame has been increased to see the effect of semi-rigid connections on fat

rhombus frame. Bi-linear moment rotation curve has been used for assigning semi-rigid

connection properties. Three different semi-rigid connection properties have been used.

Semi-rigid connections are used along all the beam ends from level 7 to level 11. Hybrid

frames with rigid and semi-rigid connections are subjected to several earthquake data,

and the change in global behavior of the frame is closely observed.

To describe the results, a general building designation is adopted for convenience,

_S – H_W_H/W- OP( _,/ _ ) – EQ – E_(B-L) , where,

S = story

H = the height of the frame

W = the width of the frame

H/W = the height to width ratio of the frame

OP = out put, which can be member forces or displacement

68

AF = axial force

SF = shear force

BM = bending moment

TDUx = top node displacement along X-direction

MDUx = maximum lateral displacement along X-direction

LDr = lateral inter story drift

RMR = Rigid Rhombus frame

RCR = Rigid Rectangular frame

RMSR = Semi-rigid Rhombus frame

RCSR = Semi-rigid Rectangular frame

EQ = earthquake

ELC = El Centro earthquake

NRG = Northridge earthquake

PKF = Parkfield earthquake

KCL1 = Afyon Bay record for Kocaeli earthquake

KCL2 = Aydin record for Kocaeli earthquake

E = the initial stiffness for the semi-rigid connections. Only the initial stiffness value has

been used in the building designations to express the semi-rigid connections

properties. Details of the connection properties with designations are given in section

4.2.3.

B = the beginning story of the range in which semi-rigid connections are placed.

L = the last story of the range in which semi-rigid connections are placed.

69

The building designation follows: 20S – 289_100_2.89 – BM(RMSR/RMR) –

NRG – E1m(7-11), which will describe a 20 story frame with height to width ratio of

2.89, where the height and the width are 289 ft and 100 ft respectively. It will express

bending moment as out put, and the result will be the ratio of bending moment between

Rhombus frame with semi-rigid connections and the Rhombus frame with all rigid

connections. The frames are analyzed under Northridge earthquake. And the last part

defines, that initial stiffness of the semi-rigid connection is 1 million, which are placed at

the beam-to-column joints from story 7 to 11.

4.2 Semi-rigid Connection

Connections in a beam-column frame are used to transfer internal forces from one

member to another. Generally, a beam-column connection assumed as either rigid or

pinned. In a rigid connection, it is assumed that the stiffness of the connection is large

enough to transfer all the moments from the beam to column to provide lateral stiffness,

and in this process, the relative rotation between the members will be zero. On the other

hand, in a pinned connection, no moment will be transferred between the connecting

members. But in the real structures, connections do not behave either of these two ideal

conditions. Many connections that are been widely used in the present day, fall between

these two conditions. When a moment is applied to a connection, there will be some

relative rotation (in case of rigid connection which is assumed to be zero) between the

adjoining members. And this type of connection is termed as semi-rigid connections.

70

4.2.1 Moment-Rotation models

Semi-rigid connection is one of the major sources of nonlinearity in a steel frame.

Generally, the axial and shear deformations are assumed to be negligible compared to the

flexural deformation. For that reason, only the flexural deformation is considered to have

effect on rotational deformation of a connection in this study. Figure 4.1 shows rotational

deformation of a connection.

Figure 4.1 Rotational deformation of a connection due to flexure

Thus, a moment versus rotation curve can express the connection behavior in a

form of, M = f(φ), where M is the connection moment, and φ is the relative rotation

between the adjoining members. Many analytical, mathematical and mixed models,

which combine analytical and mathematical models, are proposed. In a moment-rotation

curve, the basic properties are strength, stiffness and the ductility. These properties are

shown in Figure 4.2.

M

Φ

71

Figure 4.2 Typical moment-rotation curves

To include the effect of a semi-rigid connection, a connection spring can be

introduced at the joining point of a beam and column. The two ends of a beam thus

connected by the spring with the columns formed a hybrid element, which is shown in

Figure 4.3. Rotation of the connection is defined as the relative angle between the two

ends of the spring which is assumed to be a zero length element.

Figure 4.3 Semi-rigid element of a Hybrid frame

Rotation

strength

Φ

M

Mom

ent a

t con

nect

ion

duct

ility

M - Φ curve

Bi Linear Model

Trilinear Model

Tangent stiffness

Col

umn

elem

ent

Col

umn

elem

ent

Col

umn

elem

ent

Col

umn

elem

ent

Hybrid Element

Zero Length Element

Zero Length Element

Beam Element

72

4.2.2 Bi-linear semi-rigid Connection Properties

A moment-rotation curve is used as a medium to express the nonlinear behavior

of a connection. Many M- φ relations have been proposed. In this study, to simulate

semi-rigid behavior of the connection, different bilinear moment-rotation curves have

been used for simplicity, with no provision for ultimate capacity. A general bilinear

moment rotation curve is shown in Figure 4.4.

Rotation,

Mom

ent,

M

Φ

Figure 4.4 Bi-linear moment-rotation curve

Figure 4.4 shows bilinear connection model used in this study, which requires few

parameters to describe. These are initial and post yield stiffness and yield moment. In all

the cases, ratio of the post yield to initial stiffness is taken as 0.01. The initial stiffness of

the connection changes from a low value to higher one in an attempt to describe a wide

range of semi-rigid connections. Yield moment value changes with the initial stiffness.

The different parameter values are tabulated in Table 4.1.

E

0.01E My

73

Table 4.1 Semi-rigid Connection properties

Designation

Initial Stiffness

kip-in/rad

Post-Yield Stiffness

kip-in/rad

Post yield to

initial stiffness

ratio

Yield Moment

kip-in

E1m 1,000,000 10,000 0.01 1989

E3m 3,000,000 30,000 0.01 2323

E5m 5,000,000 50,000 0.01 3030

4.3 Two-Story Test Frame

For verification of the proposed modeling for semi-rigid connection, a simple

structure is required whose response is known. For this purpose a 2-Story one bay frame,

which is analyzed before using semi-rigid connection by Bhatti and Hingtgen (1995) and

King and Chen (1993) was chosen. The geometry and dimensions of that simple frame

modeled by them is shown in Figure 4.5. In this study, this frame is modeled in Opensees

software in an attempt to verify it’s capability to incorporate semi-rigid connection

property in the region of beam-column connections. Four vertically downward 100 kips

load is acting at node 3, 4, 5 and 6; and two horizontal 10 kips load is acting at node 3

and 5.

74

Figure 4.5 Two story one bay test frame

At first, the frame is analyzed assuming all the connections being rigid with no

geometric effect and the results are compared with the previous results from the papers.

In the next stage, geometric effect is considered in the form of P-Δ effects with the

connections being rigid. Finally, both the geometric nonlinearity (P-Δ) and the

connection nonlinearity are incorporated. Connection nonlinearity is due to the semi-rigid

property of the connections. Semi-rigid connections are being simulated using zero length

spring elements at the end of the beam elements. Each 2-noded spring element has two

transitional and one rotational degree of freedom at each node. But transitional degrees of

freedom are assumed identical and rotational degrees of freedom are different to simulate

relative rotation as in semi-rigid connection.

75

For moment-rotation curve, a bilinear elastic plastic moment-rotation curve has

been used. The moment-rotation curves used by King and Chen (1993) and Bhatti and

Hingtgen (1995) are shown in Figure 4.6. Present study follows the M-φ relationship

used by Bhatti and Hingtgen (1995), where they assumed initial stiffness for elastic

region with a value of 786732 kip-in/rad and Yield moment as 1989 kip-in.

Figure 4.6 Moment-Rotation relation used in previous study (Bhatti and Hingtgen,1995)

Lateral displacements at node 3 and 5, and absolute bending moments of all the elements

are tabulated in Table 4.2 and 4.3 for comparison.

Table 4.2 Lateral displacement (inch) for 2-Story test frame

Rigid without P- Δ Rigid with P- Δ Semi-rigid with P- Δ

Node #3 Node#5 Node #3 Node #5 Node #3 Node #5

Bhatti & Hingtgen 1.011 1.509 1.168 1.731 1.477 2.292

King & Chen - - 1.16 1.82 2.02 3.26

This Study 1.01184 1.51099 1.1623 1.72528 1.47056 2.28543

76

Table 4.3 Absolute Maximum Bending Moment (kip-inch) for 2-Story test frame

Element # 1 2 3 4 5 6

Rigid without P- Δ

Bhatti & Hingtgen 1450 711 1443 1437 711 711

King & Chen - - - - - - This Study 1449 711 1443 1437 711 711

Rigid with P- Δ

Bhatti & Hingtgen 1654 795 1677 1669 794 795

King & Chen 1649 794 1670 1664 794 794 This Study 1653 796 1675 1669 796 796

Semi-rigid with P- Δ

Bhatti & Hingtgen 1634 902 1739 1731 902 902

King & Chen 1560 1116 1837 1834 1116 1116 This Study 1633 903 1736 1731 903 903

The data presented in Table 4.2 and 4.3 verifies that the output from Opensees is

reliable based on the previous results. The results from the present study are much closer

with the results from Bhatti and Hingtgen (1995), as in both the cases moment-rotation

relation is similar which is a bilinear elastic plastic curve, while King and Chen (1993)

had used a different moment-rotation relation which may cause some difference.

4.4 20-Story Rectangular Hybrid Frame

Initially, a 20story frame has been modeled in Opensees software with SAC 20-

Story building dimensions. This 20-Story 5 bay frame is used in this study to observe the

effect of earthquake on high-rise buildings. The rectangular frame is 100 ft wide with 20

ft bay dimension, and the total height of the building is 289ft with two basement floor,

12ft each. All the member sizes are W24 X 131. Total dead load and 25 percentages of

live loads are considered for calculating effective seismic load. Load calculations are

77

described in Appendix G. At first, all the connections are considered to be rigid. Later,

partially restrained connections are incorporated in the structure to see the effect of semi-

rigid connections in the hybrid frames. All beam-to-column connections from level-7 to

11 are modeled as partially restrained. This configuration is selected based on the

previous study by Dobrinka Radulova (2009). The semi-rigid connections are

incorporated in the model by using a zero length spring element at the ends of the beam

members as shown in figure 4.3. Then, the semi-rigid connection property is assigned to

this spring element. Three different semi-rigid connections as per Table 4.1 have been

used. 20-story rectangular hybrid frame is shown in Figure 4.7. One rigid and three 20-

story hybrid rectangular frames are analyzed under five different earthquake excitation

records. Performance of the hybrid rectangular frame is compared with the rigid

rectangular frame and rigid rhombus frame.

78

Figure 4.7 Hybrid Rectangular: 20S – 289_100_2.89 – (RCSR) – E(7-11)

79

4.5 20-Story Rhombus Hybrid Frame

The 20-Story base rhombus frame has been modeled in Opensees software as

discussed in section 3.5. This frame is used in this study to observe the effect of semi-

rigid connections on rhombus shape buildings. To see the effect of height-to-width ratio

on the performance of the high rise structure, total width of this building is increased to

300ft. From 5, bay number increases to 15; Total Height and individual bay width remain

same. All the member sizes are W24 X 131. At the beginning all the connections are

considered to be rigid. Then, partially restrained connections are used on all the beam

ends from levels 7 though 11. The semi-rigid connections are introduced as a zero length

spring element following the previous section 4.4. Three different semi-rigid connections

as per Table 4.1 have been used. 20-story rhombus hybrid frames are shown in Figure 4.8

and 4.9. Two rigid and six 20-story hybrid rhombus frames are analyzed under five

different earthquake excitation records.

80

Figure 4.8 Hybrid Rhombus: 20S – 289_100_2.89 – (RMSR) – E(7-11)

81

Figure 4.9 Fat Hybrid Rhombus: 20S – 289_300_0.963 – (RMSR) – E(7-11)

82

4.6 Discussion of the Results

In this chapter, hybrid frames are compared with the corresponding rigid frames.

Then, the rectangular hybrid frame is compared with the rigid rhombus frame. Hybrid

rectangular frames show improved performance than the rectangular rigid frame in most

of the cases. Top node displacements for rectangular rigid and hybrid frames are shown

in Figure A.21 through A.25. Maximum displacements for each story are plotted from

Figure E.21 through E.25. Figure 4.10 shows the displacement profile for 20S –

289_100_2.89–(RCR, RCSR)–E1m (7-11) frames. It shows, for ElCentro, Northridge

and Parkfield earthquakes, maximum lateral sway decreased for hybrid rectangular frame

comparing to the rigid rectangular frame. But hybrid frame gives more lateral

displacement than rigid frame when subjected to the Kocaeli earthquakes.

-2-10123456789

101112131415161718192021

0 8 16 24 32 40Maximum Displacement, Ux (inch)

Stor

y

-18.4 1.6 21.6 41.6 61.6 81.6 101.6Maximum Displacement, Ux(cm)

Parkfield, rigidElCentro, rigidNorthridge, rigidKocaeli-1, rigidKocaeli-2,rigidParkfield, semi-rigidElCentro, semi-rigidNorthridge, semi-rigidKocaeli-1, semi-rigidKocaeli-2, semi-rigid

20S-289_100_2.89-MDux-E1m(7-11)

Figure 4.10 Maximum lateral sway 20S-289_100_2.89–MDUx (RCR,RCSR)–E1m(7-11)

83

This can be explained by comparing the frequency of the frame with the

earthquake frequencies. Earthquake frequencies are tabulated in chapter 3 from Table 3.5

through 3.10. Hybrid and rigid frame frequencies obtained from the Opensees software

are tabulated in Table 4.4.

Table 4.4 Rectangular rigid and hybrid frame frequencies (Opensees)

Mode 20S –289_100_2.89–RCR 20S-289_100_2.89-RCSR-E1m (7-11) (Hz) (Hz)

1 0.3962 0.34146 2 1.24918 1.18438 3 2.24456 1.89239 4 3.21852 2.85043 5 4.24537 3.92719 6 5.30505 4.73884 7 6.42361 5.98415 8 7.58788 7.08584 9 8.80896 8.24295

10 10.0716 9.56393

First two mode frequencies for the hybrid frame are much closer to the first two

dominant frequencies for Afyon Bay and Aydin records for Kocaeli earthquake. This

causes resonance effect for hybrid frame when subjected to Kocaeli earthquake. Earlier it

was defined that, ElCentro and Northridge are high frequency, and Parkfield is a medium

frequency earthquakes. High rise structures have low frequencies, and semi-rigid

connections reduce it more. So, high and medium frequency earthquakes do not create

resonance effect on hybrid rectangular frames unless the original rigid structure is too

stiff.

84

Inter story lateral drifts for rigid and hybrid rectangular frames are plotted in

Figure F.21 through F.25. Figure 4.11 shows the inter story drift profile for hybrid and

rigid rectangular frame for one type of semi rigid connections. For Northridge

earthquake, hybrid frame experience less drift than the rigid frame. Hybrid frame

subjected to ElCentro and Parkfield earthquake gives greater drift values in the mid levels

of the structure than the rigid rectangular frame. These increases in the lateral drift in

hybrid frame are due to the placing of semi-rigid connections at the mid levels of the

structure. Semi-rigid connections make a structure flexible and for that reason, in those

levels the structure experience more drifts than the rigid frame. Both records from

Kocaeli earthquake gives more drift values in hybrid frame than the rigid rectangular

frame due to resonance effect discussed before.

-2-10123456789

101112131415161718192021

0 0.005 0.01 0.015 0.02Lateral Drift

Sto

ry


20S-289_100_2.89-LDr-E1m(7-11)Life Safety limit 0.025Collapse Prevention limit 0.05

Figure 4.11 20S – 289_100_2.89–LDr (RCR,RCSR) –E1m(7-11)

85

Internal forces are also compared between hybrid and rigid rectangular frames. In

most of the cases, hybrid frames perform better than the rigid frames, when subjected to

high and medium frequency earthquakes. But hybrid frames do not perform as expected,

under low frequency Kocaeli earthquake. This is due to the resonance effect. Figure 4.12

shows the beam bending moment ratios from hybrid and rigid rectangular frame.

-2-10123456789

101112131415161718192021

0 1 2 3Bending Moment ratio

Sto

ry

Data for ParkfieldData for ElCentroData for NorthridgeData for Kocaeli-1Data for Kocaeli-2

20S-289_100_2.89-BM(RCSR/RCR)-E1m(7-11)

Figure 4.12 20S – 289_100_2.89 – BM (RCSR/RCR) – E1m(7-11) for Beams

Shear force and bending moment ratios in Figure C.21 thorough C.25 and D.21

through D.25 shows the similar pattern. Except low frequency Kocaeli earthquake, in

most of the cases hybrid frames demand less forces than the rigid frames. The beam

bending moment demand reduces 11 times lower than the demand from rigid rectangular

frame. None of the beams in ElCentro and Northridge earthquake demand more beam

bending force than the rigid rectangular frame. The maximum increase in beam bending

moment demand is 2 times higher than the rigid frame for Aydin record of Kocaeli

86

earthquake. Maximum column moment demand increase in hybrid frames is 2.5 times

higher than the rigid rectangular frame.

Axial force ratios between hybrid and rigid rectangular frames are shown through

Figure B.21 through B.25. Axial force demand for hybrid frames increase at the mid level

due to the placing of semi-rigid connections at these locations. Maximum increase in

beam axial force is 108 times higher than the rigid frame when subjected to Aydin record

of Kocaeli earthquake. And maximum decrease in demand is 26 times lower than the

rigid frame for Northridge earthquake. In most of the cases axial force demand is less for

hybrid rectangular frames. Only demand increase when subjected to Kocaeli records.

Highest increase is 50 percent than the rigid frame. Maximum decrease in column axial

force demand is 7 times lower than the rigid rectangular frame for Northridge earthquake.

Hybrid rhombus frame is also compared with the rigid rhombus frame. But the

effect of semi-rigid connections in the rhombus frame is not that much significant as

observed in case of rectangular frame. Frequency of rigid and hybrid frames are almost

equal. Frequencies obtained for hybrid and rigid rhombus frame modeled in Opensees

software is tabulated in table 4.5.

87

Table 4.5 Rhombus rigid and hybrid frame frequencies (Opensees)

Mode 20S –289_100_2.89–RMR 20S-289_100_2.89-RMSR-E1m (7-11)

(Hz) (Hz) 1 0.64306 0.63469 2 2.251356 1.728858 3 3.985531 3.653391 4 4.878337 4.468481 5 6.906455 6.060516 6 7.946315 7.009683 7 9.560905 8.739879 8 10.88132 9.966534 9 12.51924 11.65015

10 14.31197 13.53017

Semi-rigid connections can not provide expected flexibility to the hybrid rhombus

frame. As, both of the hybrid and rigid frame bear almost same frequency, Figure A.26 to

A.30 and E.26 to E.30 shows top node displacement and sway profile which do not

significantly varies for hybrid rhombus frames than the rigid rhombus frames. Figure

4.13 shows the lateral sway for 20S –289_100_2.89–(RMR, RMSR)–E1m (7-11) frames.

Figure F.26 through F.30 show, that the frame has more lateral drift in the mid levels for

hybrid frames than the rigid rhombus frame, as semi-rigid connections are placed in those

places. On the other levels with rigid connections, hybrid rhombus frame demand less

lateral drift than the rigid rhombus frame. Figure 4.14 shows inter story drifts for 20S –

289_100_2.89–(RMR, RMSR)–E1m (7-11) frames.

88

-2-10123456789

101112131415161718192021

0 4 8 12 16 20Maximum Displacement, Ux (inch)

Stor

y0 10 20 30 40 50

Maximum Displacement, Ux(cm)


Figure 4.13 Maximum lateral sway 20S-289_100_2.89-MDUx(RMR,RMSR)-E1m(7-11)

-2-10123456789

101112131415161718192021

0 0.005 0.01 0.015Lateral Drift

Sto

rey


20S-289_100_2.89-LDr(RMR,RMSR)-E1m(7-11)Life Safety limit 0.025Collapse Prevention limit 0.05

Figure 4.14 20S –289_100_2.89–LDr (RMR,RMSR)–E1m(7-11)

89

Hybrid rhombus frame does not show better redistribution pattern of internal

forces as hybrid rectangular frame has shown previously. Placing of semi-rigid

connections at the mid levels scattered points at that level for internal force ratios

between hybrid and rigid rhombus frames. Below and upper level of frame does not

experience significant change in demand for hybrid rhombus frame than the rigid

rhombus frame. Figure 4.15 shows the beam bending moment ratio between hybrid and

rigid frames.

-2-10123456789

101112131415161718192021


Stor

y

Data for ParkfieldData for ElCentroData for NorthridgeData for Kocaeli-1Data for Kocaeli-2

20S-289_100_2.89-BM(RCSR/RCR)-E1m(7-11)

Figure 4.15 20S – 289_100_2.89 – BM (RMSR/RMR) – E1m(7-11) for Beams

With the usage of stiffer semi-rigid connections, the internal force ratios move to

near 1. These are shown in figure B.26 to B.30, C.26 to C.30 and D.26 to D.30.

Maximum column axial force demand for hybrid rhombus frame is around 7 times higher

and minimum is 5 times lower than the rigid rhombus frame demand for Aydin record of

90

Kocaeli earthquake. Maximum beam shear demand is 192 percent higher and minimum

demand is 138 percent lower for hybrid rhombus frame than rigid rhombus frame for

Aydin record of Kocaeli earthquake. Hybrid rhombus frame demand maximum 185

percent higher moment than the rigid rhombus frame subjected to Parkfield earthquake,

and minimum beam moment demand is 138 percent lower for hybrid rhombus frame than

the rigid rhombus frame.

Internal forces for the rigid rhombus frames are compared to the hybrid

rectangular frames. Ratios of axial force demand between rigid rhombus frame and

hybrid rectangular frame are shown in figure B.31 to B.35. In most of the cases, rigid

rhombus frame demands more column axial force, than the hybrid rectangular frame. The

maximum demand of column axial force for the rigid rhombus frame is 5.5 times higher

than the demand for hybrid rectangular frame, subjected to Parkfield earthquake. The

column axial force demands on the top floors for rigid rhombus frame are significantly

lower than the hybrid rectangular frame. The minimum demand for column axial force

for rigid rhombus frame is 26 times lower than the hybrid rectangular frame for Aydin

record of Kocaeli earthquake.

Figure C.31 to C.35 and D.31 to D.35 shows the shear force and bending moment

ratios between rigid rhombus frame and hybrid rectangular frame. Rigid rhombus frame

improves building performance for low earthquake records. And the minimum bending

moment required by rigid rhombus frame is 71 times lower for column, and 24 times

91

lower for beam, than the hybrid rectangular frame, for Aydin record of Kocaeli

earthquake. Most of the beams and columns below mid level of the rigid rhombus frame,

demand more shear force and bending moment than the hybrid rectangular frame

subjected to medium and high frequency earthquakes. The maximum beam bending

moment demand for rigid rhombus frame is 3.4 times higher than the hybrid rectangular

frame bending demand, for Parkfield earthquake. Top floors above mid levels of rigid

rhombus frame demand less shear and bending moment for column and beams. Minimum

bending moment demand for rigid rhombus frame is around 12 times lower than the

demand for hybrid rectangular frame for ElCentro earthquake.

Figures B.36 to B.40, C.36 to C.40 and D.36 to D.40 show the axial force, shear

force and bending moment ratios between fat hybrid rhombus frame with the fat rigid

rhombus frame. This frames are designated by 20S –289_300_0.963–(RMR, RMSR)–(7-

11). Results from this frames show scattered point in the mid levels where semi rigid

connections are placed. On the other levels, these internal force ratios move close to one.

92

CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

5.1 Summary

The primary objective of this study is to present an efficient building geometric

shape for earthquake force resisting frame system. A rhombus shape is created from the

popular rectangular moment frame for this purpose. In the first stage, several rhombus

frames have been constructed to cover from low-rise to high-rise buildings. Then, these

frames are analyzed under five earthquake records with wide frequency contents. The

results are compared with the rectangular frames with the same geometric aspect ratios.

Then, different semi-rigid connection properties are introduced in the high rise rhombus

and rectangular frames. Behaviors of hybrid frames with both rigid and semi-rigid

connections, when subjected to earthquake excitations, are closely observed, and results

are compared with their equivalent rigid frames. The width of the high rise rhombus

frame is increased to observe the effect on seismic performance. This fat rhombus frame

is also compared with its equivalent hybrid frames. In the last stage, the rigid rhombus

frame is compared with the hybrid rectangular frame.

93

5.2 Conclusions

Comparing four rigid rhombus frames to four rigid rectangular frames under five

earthquake records, the rhombus frames performed better than the rectangular frames.

Introducing semi-rigid connections in the high-rise structures, the hybrid rectangular

frame with rigid and semi-rigid connections showed improved performance over the rigid

rectangular frame. Comparing the hybrid rectangular frame to the rigid rhombus frame,

the major conclusions drawn from this study are summarized below –

Rigid rhombus frames are much stiffer than the rigid rectangular frames.

Top node displacements and lateral sway values from the 3-, 9-, 12- and 20-story

rigid rhombus frames are less than that of the equivalent rigid rectangular frames.

For all the considered earthquakes, 3-, 9- and 12- story rigid rhombus frames

experience less inter story drift than the rectangular rigid frames. But, few stories

of a 20-story rigid rhombus frame experience a little more drift than the rigid

rectangular frame for Parkfield and Afyon Bay records for Kocaeli earthquakes.

In most of the cases, rigid rhombus frames demand less shear force and bending

moment than the rigid rectangular frames.

In few cases, bottom stories of 20-story rigid rhombus frame demand more shear

force and bending moment than the rigid rectangular frame. But, this is negligible

comparing to the reduction of demand in other stories.

Rigid rhombus frames demand more axial force than the rigid rectangular frames.

This is true for both the beams and columns. Beams in the middle portion of the

rigid rhombus frames demand maximum axial force in most of the cases.

94

Semi-rigid connections increase the flexibility of the frames, and thus reduce the

frequency of the structure. Semi-rigid connections have significant influence on

reducing the frequency of rectangular frames; but they do not have a significant

effect on rhombus frames.

20-story rectangular hybrid frames with semi-rigid connections in the mid levels

exhibited improve building performance over the equivalent rigid rectangular

frame, when subjected to medium and high frequency earthquakes. Due to

reduction of frame frequency, by use of semi-rigid connection the probability of

resonance in high and medium frequency earthquakes is lowered. This results in

less internal forces demand. But for low frequency earthquakes, high rise

rectangular frames with semi-rigid connections are good candidates to produce

resonance.

Hybrid rectangular frames exhibit more lateral drift in the stories where semi-

rigid connections are placed. Sometimes this condition could result in a soft story

mechanism. Experienced judgment is required for placing semi-rigid connections

in rectangular frames.

The location pattern used in this study for semi-rigid connections does not have

much effect on the rhombus shape hybrid frame. Frame frequency reduces

slightly. The frame does show some flexibility on the stories, where semi-rigid

connections are placed, but the effect on the global behavior of the frame is

insignificant.

95

A fat hybrid rhombus frame behaves in the same way as a base hybrid rhombus

frame.

Redistribution of internal forces depends on the semi-rigid connection property.

Hybrid frames with relatively stiffer semi-rigid connections, demand internal

forces almost same as that for rigid frames.

High rise rigid rhombus frames perform better than the high rise hybrid

rectangular frames in low frequency earthquakes.

5.3 Recommendations

As this study attempts to propose a new earthquake resistance framing system, the

following recommendations for future work are made –

Semi-rigid connections need to be applied at different locations in the rhombus

frames to observe the seismic performance. Here in this study, semi-rigid

connections are placed only on the mid stories. Different pattern of placing semi-

rigid connections may significantly modify the results for rhombus shape.

Modifications in the frame assembly can be tried to reduce the axial force

demand in Rhombus frame.

Comparisons to be made with other bracing systems such as concentrically

braced frame and eccentrically braced frame.

96

APPENDIX A

TOP LATERAL DISPLACEMENT

97

-8

-6

-4

-2

0

2

4

6

8

0 5 10 15 20 25 30 35 40Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-20.32

-16.32

-12.32

-8.32

-4.32

-0.32

3.68

7.68

11.68

15.68

19.68

Dis

plac

emen

t-Ux(

cm)

3S-39_120_0.325-TDUx(RMR)-PKF 3S-39_120_0.325-TDUx(RCR)-PKF

Figure A.1 Top lateral displacement for 3S-39_120_0.325-PKF

-4

-3

-2

-1

0

1

2

3

4

0 5 10 15 20 25 30Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-10.16

-7.66

-5.16

-2.66

-0.16

2.34

4.84

7.34

9.84

Dis

plac

emen

t-Ux(

cm)

3S-39_120_0.325-TDUx(RMR)-ELC 3S-39_120_0.325-TDUx(RCR)-ELC

Figure A.2 Top lateral displacement for 3S-39_120_0.325-ELC

98

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 10 20 30 40 50Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-12.7

-10.2

-7.7

-5.2

-2.7

-0.2

2.3

4.8

7.3

9.8

12.3

Dis

plac

emen

t-Ux(

cm)

3S-39_120_0.325-TDUx(RMR)-NRG 3S-39_120_0.325-TDUx(RCR)-NRG

Figure A.3 Top lateral displacement for 3S-39_120_0.325-NRG

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 30 60 90 120 150 180Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-0.381

-0.301

-0.221

-0.141

-0.061

0.019

0.099

0.179

0.259

0.339

Dis

plac

emen

t-Ux(

cm)

3S-39_120_0.325-TDUx(RMR)-KCL1 3S-39_120_0.325-TDUx(RCR)-KCL1

Figure A.4 Top lateral displacement for 3S-39_120_0.325-KCL1

99

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 30 60 90 120 150 180 210Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-0.127

-0.102

-0.077

-0.052

-0.027

-0.002

0.023

0.048

0.073

0.098

0.123

Dis

plac

emen

t-Ux(

cm)



-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-50.8

-40.8

-30.8

-20.8

-10.8

-0.8

9.2

19.2

29.2

39.2

49.2

Dis

plac

emen

t-Ux(

cm)



100

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 5 10 15 20 25 30Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-25.4

-20.4

-15.4

-10.4

-5.4

-0.4

4.6

9.6

14.6

19.6

24.6

Dis

plac

emen

t-Ux(

cm)



-25

-20

-15

-10

-5

0

5

10

15

20

25

0 10 20 30 40 50Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-63.5

-53.5

-43.5

-33.5

-23.5

-13.5

-3.5

6.5

16.5

26.5

36.5

46.5

56.5

Dis

plac

emen

t-Ux(

cm)



101

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 30 60 90 120 150 180Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-1.524

-1.224

-0.924

-0.624

-0.324

-0.024

0.276

0.576

0.876

1.176

1.476

Dis

plac

emen

t-Ux(

cm)



-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180 210Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-2.54

-2.04

-1.54

-1.04

-0.54

-0.04

0.46

0.96

1.46

1.96

2.46

Dis

plac

emen

t-Ux(

cm)



102

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-50.8

-40.8

-30.8

-20.8

-10.8

-0.8

9.2

19.2

29.2

39.2

49.2

Dis

plac

emen

t-Ux(

cm)



-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-50.8

-40.8

-30.8

-20.8

-10.8

-0.8

9.2

19.2

29.2

39.2

49.2

Dis

plac

emen

t-Ux(

cm)



103

-40

-30

-20

-10

0

10

20

30

40

0 10 20 30 40 50Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-101.6

-81.6

-61.6

-41.6

-21.6

-1.6

18.4

38.4

58.4

78.4

98.4

Dis

plac

emen

t-Ux(

cm)



-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-2.54

-2.04

-1.54

-1.04

-0.54

-0.04

0.46

0.96

1.46

1.96

2.46

Dis

plac

emen

t-Ux(

cm)



104

-1.5

-1.2

-0.9

-0.6

-0.3

0

0.3

0.6

0.9

1.2

1.5

0 30 60 90 120 150 180 210Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-3.81

-3.01

-2.21

-1.41

-0.61

0.19

0.99

1.79

2.59

3.39

Dis

plac

emen

t-Ux(

cm)



-25

-20

-15

-10

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-63.5

-53.5

-43.5

-33.5

-23.5

-13.5

-3.5

6.5

16.5

26.5

36.5

46.5

56.5

Dis

plac

emen

t-Ux(

cm)



105

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-50.8

-40.8

-30.8

-20.8

-10.8

-0.8

9.2

19.2

29.2

39.2

49.2

Dis

plac

emen

t-Ux(

cm)



-40

-30

-20

-10

0

10

20

30

40

0 10 20 30 40 50Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-101.6

-81.6

-61.6

-41.6

-21.6

-1.6

18.4

38.4

58.4

78.4

98.4

Dis

plac

emen

t-Ux(

cm)



106

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-2.54

-2.04

-1.54

-1.04

-0.54

-0.04

0.46

0.96

1.46

1.96

2.46

Dis

plac

emen

t-Ux(

cm)



-1.75

-1.45

-1.15

-0.85

-0.55

-0.25

0.05

0.35

0.65

0.95

1.25

1.55

0 30 60 90 120 150 180 210Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-4.445

-3.645

-2.845

-2.045

-1.245

-0.445

0.355

1.155

1.955

2.755

3.555

4.355

Dis

plac

emen

t-Ux(

cm)



107

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-63.5

-53.5

-43.5

-33.5

-23.5

-13.5

-3.5

6.5

16.5

26.5

36.5

46.5

56.5

Dis

plac

emen

t-Ux(

cm)

RigidSemi-rigid, E=1millionSemi-rigid, E=3millionSemi-rigid, E=5million

20S-289_100_2.89-TDux-PKF-E(7-11)


-25

-20

-15

-10

-5

0

5

10

15

20

25

0 5 10 15 20 25 30

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-63.5

-53.5

-43.5

-33.5

-23.5

-13.5

-3.5

6.5

16.5

26.5

36.5

46.5

56.5

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-ELC-E(7-11)


108

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 10 20 30 40 50

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-127

-107

-87

-67

-47

-27

-7

13

33

53

73

93

113

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-NRG-E(7-11)


-1.5

-1.2

-0.9

-0.6

-0.3

0

0.3

0.6

0.9

1.2

1.5

0 36 72 108 144 180

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-3.81

-3.01

-2.21

-1.41

-0.61

0.19

0.99

1.79

2.59

3.39

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-KCL1-E(7-11)


109

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

1.75

0 40 80 120 160 200

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-4.445

-3.545

-2.645

-1.745

-0.845

0.055

0.955

1.855

2.755

3.655

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-KCL2-E(7-11)


-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-50.8

-40.8

-30.8

-20.8

-10.8

-0.8

9.2

19.2

29.2

39.2

49.2

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-PKF-E(7-11)


110

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 5 10 15 20 25 30

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-63.5

-53.5

-43.5

-33.5

-23.5

-13.5

-3.5

6.5

16.5

26.5

36.5

46.5

56.5

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-ELC-E(7-11)


-20

-15

-10

-5

0

5

10

15

20

0 10 20 30 40 50

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-50.8

-40.8

-30.8

-20.8

-10.8

-0.8

9.2

19.2

29.2

39.2

49.2

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-NRG-E(7-11)


111

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-2.54

-2.04

-1.54

-1.04

-0.54

-0.04

0.46

0.96

1.46

1.96

2.46

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-KCL1-E(7-11)


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180 210

Time(sec)

Dis

plac

emen

t-Ux(

inch

)

-2.54

-2.04

-1.54

-1.04

-0.54

-0.04

0.46

0.96

1.46

1.96

2.46

Dis

plac

emen

t-Ux(

cm)


20S-289_100_2.89-TDux-KCL2-E(7-11)


112

APPENDIX B

AXIAL FORCES IN THE FRAME ELEMENTS

113

0

1

2

3

0 1 2 3 4 5Axial Force ratio

Stor

y

Absolute AFmax ratio 3S-39_120_0.325-AF(RMR/RCR)-PKF

a)

0

1

2

3

4


Stor

y


b)

Figure B.1 Axial Force ratio for 3S-39_120_0.325-PKF a)Beams, b)Columns

114

0

1

2

3


Stor

y

Absolute AFmax ratio 3S-39_120_0.325-AF(RMR/RCR)-ELC

a)

0

1

2

3

4


Stor

y


b)

Figure B.2 Axial Force ratio for 3S-39_120_0.325-ELC a)Beams, b)Columns

115

0

1

2

3


Stor

y

Absolute AFmax ratio 3S-39_120_0.325-AF(RMR/RCR)-NRG

a)

0

1

2

3

4


Stor

y


b)

Figure B.3 Axial Force ratio for 3S-39_120_0.325-NRG a)Beams, b)Columns

116

0

1

2

3


Stor

y

Absolute AFmax ratio 3S-39_120_0.325-AF(RMR/RCR)-KCL1

a)

0

1

2

3

4


Stor

y


b)

Figure B.4 Axial Force ratio for 3S-39_120_0.325-KCL1 a)Beams, b)Columns

117

0

1

2

3


Stor

y


a)

0

1

2

3

4


Stor

y


b)


118

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


119

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


120

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


121

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


122

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


123

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


124

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

0 1 2 3 4 5 6Axial Force ratio

Stor

y


b)


125

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y

Absolute AFmax ratio12S-173_150_1.153-AF(RMR/RCR)-NRG

a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


126

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


127

0

1

2

3

4

5

6

7

8

9

10

11

12


Sto

ry


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Sto

ry


b)


128

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


129

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


130

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


131

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


132

-10123456789

1011121314151617181920


Sto

ry


a)

-10123456789

101112131415161718192021


Sto

ry


b)


133

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021

0 1 2Axial Force ratio

Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-PKF-E(7-11)

b)


134

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-ELC-E(7-11)

b)


135

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-NRG-E(7-11)

b)


136

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-KCL1-E(7-11)

b)


137

-2-10123456789

101112131415161718192021

0 30 60 90 120Axial Force ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RCSR/RCR)-KCL2-E(7-11)

b)


138

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-PKF-E(7-11)

b)


139

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-ELC-E(7-11)

b)


140

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7Axial Force ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-NRG-E(7-11)

b)


141

-2-10123456789

101112131415161718192021

0 5 10 15Axial Force ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-KCL1-E(7-11)

b)


142

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7Axial Force ratio

Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-AF(RMSR/RMR)-KCL2-E(7-11)

b)


143

-10123456789

1011121314151617181920


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-PKF-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-PKF-E1m(7-11)

b)


144

-10123456789

1011121314151617181920


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-ELC-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-ELC-E1m(7-11)

b)


145

-10123456789

1011121314151617181920


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-NRG-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-NRG-E1m(7-11)

b)


146

-10123456789

1011121314151617181920


Stor

y

Absolute AFmax ratio 20S-289_100_2.89-AF(RMR/RCSR)-KCL1-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y


b)


147

-10123456789

1011121314151617181920


Sto

ry


a)

-10123456789

101112131415161718192021


Sto

ry


b)


148

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-PKF-E(7-11)

b)


149

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1millionE=3millionE=5million

20S-289_300_0.963-AF(RMSR/RMR)-ELC-E(7-11)

b)


150

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7 8 9 10Axial Force ratio

Sto

ry

E=1millionE=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y


E=5million

20S-289_300_0.963-AF(RMSR/RMR)-NRG-(7-11)

b)


151

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-KCL1-E(7-11)

b)


152

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-AF(RMSR/RMR)-KCL2-E(7-11)

b)


153

APPENDIX C

SHEAR FORCES IN THE FRAME ELEMENTS

154

0

1

2

3

0 0.2 0.4 0.6 0.8 1Shear Force ratio

Stor

y

Absolute SFmax ratio 3S-39_120_0.325-SF(RMR/RCR)-PKF

a)

0

1

2

3

4


Stor

y


b)

Figure C.1 Shear Force ratio for 3S-39_120_0.325-PKF a)Beams, b)Columns

155

0

1

2

3


Stor

y

Absolute SFmax ratio 3S-39_120_0.325-SF(RMR/RCR)-ELC

a)

0

1

2

3

4


Stor

y


b)

Figure C.2 Shear Force ratio for 3S-39_120_0.325-ELC a)Beams, b)Columns

156

0

1

2

3


Stor

y

Absolute SFmax ratio 3S-39_120_0.325-SF(RMR/RCR)-NRG

a)

0

1

2

3

4


Stor

y


b)

Figure C.3 Shear Force ratio for 3S-39_120_0.325-NRG a)Beams, b)Columns

157

0

1

2

3


Stor

y

Absolute SFmax ratio 3S-39_120_0.325-SF(RMR/RCR)-KCL1

a)

0

1

2

3

4


Stor

y


b)

Figure C.4 Shear Force ratio for 3S-39_120_0.325-KCL1 a)Beams, b)Columns

158

0

1

2

3


Stor

y


a)

0

1

2

3

4


Stor

y


b)


159

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


160

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

0 0.3 0.6 0.9 1.2 1.5Shear Force ratio

Stor

y


b)


161

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


162

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


163

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


164

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.5 1 1.5 2 2.5Shear Force ratio

Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


165

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.5 1 1.5 2Shear Force ratio

Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


166

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


167

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


168

0

1

2

3

4

5

6

7

8

9

10

11

12


Sto

ry


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Sto

ry


b)


169

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


170

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021

0 0.5 1 1.5Shear Force ratio

Stor

y


b)


171

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


172

-10123456789

1011121314151617181920

0 1 2 3Shear Force ratio

Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


173

-10123456789

1011121314151617181920


Sto

ry


a)

-10123456789

101112131415161718192021


Sto

ry


b)


174

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021

0 1 2Shear Force ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-PKF-E(7-11)

b)


175

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-ELC-E(7-11)

b)


176

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-NRG-E(7-11)

b)


177

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-KCL1-E(7-11)

b)


178

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RCSR/RCR)-KCL2-E(7-11)

b)


179

-2-10123456789

101112131415161718192021

0 1 2 3 4 5Shear Force ratio

Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-PKF-E(7-11)

b)


180

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-ELC-E(7-11)

b)


181

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-NRG-E(7-11)

b)


182

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-KCL1-E(7-11)

b)


183

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-SF(RMSR/RMR)-KCL2-E(7-11)

b)


184

-10123456789

1011121314151617181920


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-PKF-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-PKF-E1m(7-11)

b)


185

-10123456789

1011121314151617181920


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-ELC-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-ELC-E1m(7-11)

b)


186

-10123456789

1011121314151617181920


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-NRG-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-NRG-E1m(7-11)

b)


187

-10123456789

1011121314151617181920


Stor

y

Absolute SFmax ratio 20S-289_100_2.89-SF(RMR/RCSR)-KCL1-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y


b)


188

-10123456789

1011121314151617181920


Sto

ry


a)

-10123456789

101112131415161718192021


Sto

ry


b)


189

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-PKF-E(7-11)

b)


190

-2-10123456789

101112131415161718192021


Stor

y

E=1millionE=3millionE=5million

20S-289_300_0.963-SF(RMSR/RMR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-ELC-E(7-11)

b)


191

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7 8Shear Force ratio

Stor

y


E=5million

20S-289_300_0.963-SF(RMSR/RMR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7 8Shear Force ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-NRG-(7-11)

b)

Figure C.38 Shear Force ratio for20S-289_300_0.963-NRG a)Beams, b)Columns

192

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7Shear Force ratio

Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7Shear Force ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-KCL1-E(7-11)

b)

Figure C.39 Shear Force ratio for20S-289_300_0.963-KCL1 a)Beams, b)Columns

193

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6Shear Force ratio

Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-SF(RMSR/RMR)-KCL2-E(7-11)

b)


194

APPENDIX D

BENDING MOMENTS IN THE FRAME ELEMENTS

195

0

1

2

3

0 0.2 0.4 0.6 0.8 1Bending Moment ratio

Stor

y

Absolute BMmax ratio 3S-39_120_0.325-BM(RMR/RCR)-PKF

a)

0

1

2

3

4


Stor

y


b)

Figure D.1 Bending Moment ratio for 3S-39_120_0.325-PKF a)Beams, b)Columns

196

0

1

2

3


Stor

y

Absolute BMmax ratio 3S-39_120_0.325-BM(RMR/RCR)-ELC

a)

0

1

2

3

4


Stor

y


b)

Figure D.2 Bending Moment ratio for 3S-39_120_0.325-ELC a)Beams, b)Columns

197

0

1

2

3


Stor

y

Absolute BMmax ratio 3S-39_120_0.325-BM(RMR/RCR)-NRG

a)

0

1

2

3

4


Stor

y


b)

Figure D.3 Bending Moment ratio for 3S-39_120_0.325-NRG a)Beams, b)Columns

198

0

1

2

3


Stor

y

Absolute BMmax ratio 3S-39_120_0.325-BM(RMR/RCR)-KCL1

a)

0

1

2

3

4


Stor

y


b)

Figure D.4 Bending Moment ratio for 3S-39_120_0.325-KCL1 a)Beams, b)Columns

199

0

1

2

3


Stor

y


a)

0

1

2

3

4


Stor

y


b)


200

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


201

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

0 0.3 0.6 0.9 1.2 1.5Bending Moment ratio

Stor

y


b)


202

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


203

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


204

0

1

2

3

4

5

6

7

8

9


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


205

0

1

2

3

4

5

6

7

8

9

10

11

12

0 1 2Bending Moment ratio

Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


206

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Stor

y


b)


207

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10


Stor

y


b)


208

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Sto

ry


b)


209

0

1

2

3

4

5

6

7

8

9

10

11

12


Sto

ry


a)

0

1

2

3

4

5

6

7

8

9

10

11

12

13


Sto

ry


b)


210

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021

0 1 2 3 4 5 6Bending Moment ratio

Stor

y


b)


211

0123456789

1011121314151617181920

0 0.5 1 1.5Bending Moment ratio

Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


212

-10123456789

1011121314151617181920


Stor

y


a)

-10123456789

101112131415161718192021


Stor

y


b)


213

-10123456789

1011121314151617181920

0 0.5 1 1.5 2 2.5Bending Moment ratio

Stor

y


a)

-10123456789

101112131415161718192021


Sto

ry


b)


214

-10123456789

1011121314151617181920


Sto

ry


a)

-10123456789

101112131415161718192021

0 0.5 1 1.5 2 2.5Bending Moment ratio

Sto

ry


b)


215

-2-10123456789

101112131415161718192021

0 0.5 1 1.5 2Bending Moment ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-PKF-E(7-11)

b)


216

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-ELC-E(7-11)

b)


217

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-NRG-E(7-11)

b)


218

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-KCL1-E(7-11)

b)


219

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RCSR/RCR)-KCL2-E(7-11)

b)


220

-2-10123456789

101112131415161718192021

0 1 2 3 4 5Bending Moment ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-PKF-E(7-11)

b)


221

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-ELC-E(7-11)

b)


222

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-NRG-E(7-11)

b)


223

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-KCL1-E(7-11)

b)


224

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_100_2.89-BM(RMSR/RMR)-KCL2-E(7-11)

b)


225

-10123456789

1011121314151617181920


Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-PKF-E1m(7-11)

a)

-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7Bending Moment ratio

Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-PKF-E1m(7-11)

b)


226

0123456789

1011121314151617181920


Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-ELC-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-ELC-E1m(7-11)

b)


227

-10123456789

1011121314151617181920


Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-NRG-E1m(7-11)

a)

-10123456789

101112131415161718192021


Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-NRG-E1m(7-11)

b)


228

-10123456789

1011121314151617181920


Stor

y

Absolute BMmax ratio 20S-289_100_2.89-BM(RMR/RCSR)-KCL1-E1m(7-11)

a)

-10123456789

101112131415161718192021


Sto

ry


b)


229

-10123456789

1011121314151617181920


Sto

ry


a)

-10123456789

101112131415161718192021


Sto

ry


b)


230

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-PKF-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-PKF-E(7-11)

b)


231

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-ELC-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y


E=5million

20S-289_300_0.963-BM(RMSR/RMR)-ELC-E(7-11)

b)


232

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7 8Bending Moment ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-NRG-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y


E=5million

20S-289_300_0.963-BM(RMSR/RMR)-NRG-E(7-11)

b)


233

-2-10123456789

101112131415161718192021

0 1 2 3 4 5 6 7Bending Moment ratio

Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-KCL1-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-KCL1-E(7-11)

b)


234

-2-10123456789

101112131415161718192021


Sto

ry

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-KCL2-E(7-11)

a)

-2-10123456789

101112131415161718192021


Stor

y

E=1million

E=3million

E=5million

20S-289_300_0.963-BM(RMSR/RMR)-KCL2-E(7-11)

b)


235

APPENDIX E

MAXIMUM LATERAL DISPLACEMENT PROFILE

236

0

1

2

3

4

0 2 4 6 8Maximum Displacement-Ux(inch)

Stor

y

0 5 10 15 20Maximum Displacement-Ux(cm)

3S-39_120_0.325-MDUx(RMR)-PKF 3S-39_120_0.325-MDUx(RCR)-PKF

.

Figure E.1 Maximum lateral displacement for 3S-39_120_0.325-PKF

0

1

2

3

4


Stor

y

0 2.5 5 7.5 10Maximum Displacement-Ux(cm)

3S-39_120_0.325-MDUx(RMR)-ELC 3S-39_120_0.325-MDUx(RCR)-ELC

.

Figure E.2 Maximum lateral displacement for 3S-39_120_0.325-ELC

237

0

1

2

3

4

0 1 2 3 4 5Maximum Displacement-Ux(inch)

Stor

y

0 2.5 5 7.5 10 12.5Maximum Displacement-Ux(cm)

3S-39_120_0.325-MDUx(RMR)-NRG 3S-39_120_0.325-MDUx(RCR)-NRG

.

Figure E.3 Maximum lateral displacement for 3S-39_120_0.325-NRG

0

1

2

3

4

0 0.03 0.06 0.09 0.12 0.15Maximum Displacement-Ux(inch)

Stor

y

0 0.07 0.14 0.21 0.28 0.35Maximum Displacement-Ux(cm)

3S-39_120_0.325-MDUx(RMR)-KCL1 3S-39_120_0.325-MDUx(RCR)-KCL1

.

Figure E.4 Maximum lateral displacement for 3S-39_120_0.325-KCL1

238

0

1

2

3

4


Stor

y

0 0.03 0.06 0.09 0.12Maximum Displacement-Ux(cm)


.


0

1

2

3

4

5

6

7

8

9

0 2 4 6 8 10 12 14 16 18 20Maximum Displacement-Ux(inch)

Stor

y

0 5 10 15 20 25 30 35 40 45 50

Maximum Displacement-Ux(cm)



239

0

1

2

3

4

5

6

7

8

9


Stor

y

0 5 10 15 20 25

Maximum Displacement-Ux(cm)



0

1

2

3

4

5

6

7

8

9


Stor

y

0 12 24 36 48 60Maximum Displacement-Ux(cm)


.


240

0

1

2

3

4

5

6

7

8

9


Stor

y



.


0

1

2

3

4

5

6

7

8

9


Stor

y



.


241

-1

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y




-1

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y




242

-1

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y



.


-1

0

1

2

3

4

5

6

7

8

9

10

11

12


Stor

y



.


243

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.25 0.5 0.75 1 1.25Maximum Displacement-Ux(inch)

Stor

y

0 0.6 1.2 1.8 2.4 3Maximum Displacement-Ux(cm)


.


-2-10123456789

1011121314151617181920


Stor

y

0 10 20 30 40 50 60Maximum Displacement-Ux(cm)



244

-2-10123456789

1011121314151617181920

0 3 6 9 12 15 18Maximum Displacement-Ux(inch)

Stor

y

0 7 14 21 28 35 42 49Maximum Displacement-Ux(cm)



-2-10123456789

1011121314151617181920


Stor

y



.


245

-2-10123456789

1011121314151617181920

0 0.2 0.4 0.6 0.8 1Maximum Displacement-Ux(inch)

Stor

y

0 0.5 1 1.5 2 2.5Maximum Displacement-Ux(cm)


.


-2-10123456789

1011121314151617181920

0 0.25 0.5 0.75 1 1.25 1.5 1.75Maximum Displacement-Ux(inch)

Stor

y



.


246

-2-10123456789

1011121314151617181920


Sto

ry

0 10 20 30 40 50 60Maximum Displacement-Ux(cm)

Rigid

Semi-rigid, E=1million

Semi-rigid, E=3millionSemi-rigid, E=5million

20S-289_100_2.89-MDux-PKF-E(7-11)


-2-10123456789

1011121314151617181920


Sto

ry


Rigid



20S-289_100_2.89-MDux-ELC-E(7-11)


247

-2-10123456789

1011121314151617181920

0 5 10 15 20 25 30 35 40Maximum Displacement-Ux(inch)

Sto

ry



20S-289_100_2.89-MDux-NRG-E(7-11)


-2-10123456789

1011121314151617181920

0 0.25 0.5 0.75 1 1.25Maximum Displacement-Ux(inch)

Sto

ry

0 0.5 1 1.5 2 2.5 3Maximum Displacement-Ux(cm)

Rigid



20S-289_100_2.89-MDux-KCL1-E(7-11)


248

-2-10123456789

1011121314151617181920

0 0.25 0.5 0.75 1 1.25 1.5 1.75Maximum Displacement-Ux(inch)

Sto

ry


Rigid



20S-289_100_2.89-MDux-KCL2-E(7-11)


-2-10123456789

1011121314151617181920


Sto

ry


Rigid



20S-289_100_2.89-MDux-PKF-E(7-11)


249

-2-10123456789

1011121314151617181920


Sto

ry

0 5 10 15 20Maximum Displacement-Ux(cm)

Rigid



20S-289_100_2.89-MDux-ELC-E(7-11)


-2-10123456789

1011121314151617181920

0 2 4 6 8 10 12 14 16Maximum Displacement-Ux(inch)

Sto

ry

0 5 10 15 20 25 30 35 40Maximum Displacement-Ux(cm)

Rigid



20S-289_100_2.89-MDux-NRG-E(7-11)


250

-2-10123456789

1011121314151617181920


Sto

ry


Rigid



20S-289_100_2.89-MDux-KCL1-E(7-11)


-2-10123456789

1011121314151617181920


Sto

ry


Rigid



20S-289_100_2.89-MDux-KCL2-E(7-11)


251

APPENDIX F

LATERAL INTER STORY DRIFT

252

0

1

2

3

4

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry

Lateral Drift for Rhombus Frame Lateral Drift for Rectangular Frame

.

Figure F.1 Inter Story Drift for 3S-39_120_0.325-PKF

0

1

2

3

4

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.

Figure F.2 Inter Story Drift for 3S-39_120_0.325-ELC

253

0

1

2

3

4

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.

Figure F.3 Inter Story Drift for 3S-39_120_0.325-NRG

0

1

2

3

4

0 0.0001 0.0002 0.0003 0.0004 0.0005Lateral Drift

Sto

ry


.

Figure F.4 Inter Story Drift for 3S-39_120_0.325-KCL1

254

0

1

2

3

4

0 0.00004 0.00008 0.00012 0.00016 0.0002Lateral Drift

Sto

ry


.


0

1

2

3

4

5

6

7

8

9

0 0.005 0.01 0.015 0.02 0.025 0.03Lateral Drift

Sto

ry


.


255

0

1

2

3

4

5

6

7

8

9

0 0.002 0.004 0.006 0.008 0.01Lateral Drift

Sto

ry


.


0

1

2

3

4

5

6

7

8

9

0 0.005 0.01 0.015 0.02 0.025 0.03Lateral Drift

Sto

ry


.


256

0

1

2

3

4

5

6

7

8

9

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.


0

1

2

3

4

5

6

7

8

9

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.


257

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.01 0.02 0.03Lateral Drift

Sto

ry


.


0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.01 0.02 0.03Lateral Drift

Sto

ry


.


258

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.01 0.02 0.03 0.04Lateral Drift

Sto

ry


.


0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.


259

0

1

2

3

4

5

6

7

8

9

10

11

12

0 0.0003 0.0006 0.0009 0.0012Lateral Drift

Sto

ry


.


0123456789

1011121314151617181920

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.


260

0123456789

1011121314151617181920

0 0.003 0.006 0.009 0.012 0.015Lateral Drift

Sto

ry


.


0123456789

1011121314151617181920

0 0.005 0.01 0.015 0.02 0.025Lateral Drift

Sto

ry


.


261

0123456789

1011121314151617181920

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.


0123456789

1011121314151617181920

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.


262

0123456789

1011121314151617181920

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry

RigidSemi-rigid,E=1millionSemi-rigid, E=3millionSemi-rigid, E=5million

.

20S-289_100_2.89-LDr-PKF-E(7-11)


0123456789

1011121314151617181920

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-ELC-E(7-11)


263

0123456789

1011121314151617181920

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-NRG-E(7-11)


0123456789

1011121314151617181920

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-KCL1-E(7-11)


264

0123456789

1011121314151617181920

0 0.0002 0.0004 0.0006 0.0008 0.001Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-KCL2-E(7-11)


0123456789

1011121314151617181920

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-PKF-E(7-11)


265

0123456789

1011121314151617181920

0 0.002 0.004 0.006 0.008 0.01Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-ELC-E(7-11)


0123456789

1011121314151617181920

0 0.004 0.008 0.012 0.016 0.02Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-NRG-E(7-11)


266

0123456789

1011121314151617181920

0 0.0002 0.0004 0.0006 0.0008Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-KCL1-E(7-11)


0123456789

1011121314151617181920

0 0.0002 0.0004 0.0006 0.0008Lateral Drift

Sto

ry


.

20S-289_100_2.89-LDr-KCL2-E(7-11)


267

APPENDIX G

LOAD CALCULATION

268

Load calculations for this study follow ASCE (American Society of Civil

Engineers) standards. For dead load, a 4 inch slab thickness is considered with a super

imposed dead load of 30 psf. Other dead loads are considered as per Table C3-1, ASCE

7-05.

Self weight of slab = 50 psf

Super imposed dead load = 40 psf

Mechanical duct allowance = 4 psf (Table C3-1, ASCE 7-05)

Plaster = 5 psf (Table C3-1, ASCE 7-05)

Floor finish = 32 psf (Table C3-1, ASCE 7-05)

Exterior stud walls = 48 psf (Table C3-1, ASCE 7-05)

i.e. Total dead load = 179 ≈ 180 psf

For live load calculation, residential building is considered. As per Section 4.2.2

of the ASCE 7-05, 30 psf live load for partition wall is considered. Minimum distributed

live load for residential building is 40 psf as per Table 4-1, ASCE 7-05.

i.e. Total live load = 70 psf.

For effective seismic load, total dead load and 25 percentage of live load are

considered as per Section 12.7.2, ASCE 7-05.

269

i.e. total effective seismic load = 180 + 0.25 X 70 = 197.5 psf. This load will contribute to

the mass source.

For, 20- story three dimensional building; each slab dimension is 20 ft by 20 ft.

For each beam, tributary area is equal to 100 ft2, which leads to a uniformly distributed

load on all beams = ftkip /9875.0100020

1005.197=

×× . Other tributary area will contribute to

the point load at beam ends.

Point loads for outer nodes = kip875.92

209875.0=

×

Point loads for inner nodes = 19.75 kip

For 3-, 9-, 12- story three dimensional buildings, each slab dimension is 30 ft by

30 ft. For each beam, tributary area is equal to 225 ft2, which leads to an uniformly

distributed load = ftkip /481.1100020

2255.197=

×× .

Point loads for outer nodes = kip215.222

30481.1=

×

Point loads for inner nodes = 44.43 kip

270

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BIOGRAPHICAL INFORMATION

S. M. Ashfaqul Hoq was born on 23rd May, 1984 in Dhaka, Bangladesh. He

received his Bachelor’s degree in Civil Engineering from Bangladesh University of

Engineering and Technology, Dhaka, Bangladesh in June 2007. After his graduation, he

started working in “Dalan-Kotha Limited”, a construction firm. He worked as a Project

Engineer from July 2007 to June 2008. His job responsibilities include supervision and

management of different construction projects, mainly for apartment complex. Later on,

Hoq started his graduate studies at the University of Texas at Arlington in Fall 2008. He

was awarded with the prestigious Graduate Dean Masters Fellowship scholarship. The

author worked under the guidance of Dr. Ali Abolmaali to propose a frame shape, which

will improve building performance under earthquake excitations. His Masters thesis is

substantially based on this research work.

effect of frame shape and geometry on the global by …

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